Properties

Label 32-252e16-1.1-c1e16-0-1
Degree $32$
Conductor $2.645\times 10^{38}$
Sign $1$
Analytic cond. $72250.7$
Root an. cond. $1.41853$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 3·13-s − 9·17-s − 32·25-s − 3·27-s + 6·29-s + 6·31-s + 37-s + 6·41-s − 2·43-s − 18·47-s + 7·49-s − 15·59-s + 3·61-s − 7·67-s − 79-s − 21·89-s − 3·91-s + 3·97-s + 48·101-s + 87·107-s − 14·109-s − 9·113-s + 9·119-s + 86·121-s − 24·125-s + 127-s + ⋯
L(s)  = 1  − 0.377·7-s + 0.832·13-s − 2.18·17-s − 6.39·25-s − 0.577·27-s + 1.11·29-s + 1.07·31-s + 0.164·37-s + 0.937·41-s − 0.304·43-s − 2.62·47-s + 49-s − 1.95·59-s + 0.384·61-s − 0.855·67-s − 0.112·79-s − 2.22·89-s − 0.314·91-s + 0.304·97-s + 4.77·101-s + 8.41·107-s − 1.34·109-s − 0.846·113-s + 0.825·119-s + 7.81·121-s − 2.14·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{32} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(72250.7\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{252} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.753064848\)
\(L(\frac12)\) \(\approx\) \(1.753064848\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + p T^{3} - p^{3} T^{5} + p^{3} T^{6} + p^{2} T^{7} - 8 p^{2} T^{8} + p^{3} T^{9} + p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{13} + p^{8} T^{16} \)
7 \( 1 + T - 6 T^{2} - 16 T^{3} - 10 T^{4} + 39 T^{5} + 55 p T^{6} - 239 T^{7} - 2880 T^{8} - 239 p T^{9} + 55 p^{3} T^{10} + 39 p^{3} T^{11} - 10 p^{4} T^{12} - 16 p^{5} T^{13} - 6 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
good5 \( ( 1 + 16 T^{2} + 12 T^{3} + 151 T^{4} + 33 p T^{5} + 1096 T^{6} + 1236 T^{7} + 6142 T^{8} + 1236 p T^{9} + 1096 p^{2} T^{10} + 33 p^{4} T^{11} + 151 p^{4} T^{12} + 12 p^{5} T^{13} + 16 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
11 \( 1 - 86 T^{2} + 3801 T^{4} - 115279 T^{6} + 2684372 T^{8} - 50825121 T^{10} + 73526921 p T^{12} - 11028714095 T^{14} + 130183029216 T^{16} - 11028714095 p^{2} T^{18} + 73526921 p^{5} T^{20} - 50825121 p^{6} T^{22} + 2684372 p^{8} T^{24} - 115279 p^{10} T^{26} + 3801 p^{12} T^{28} - 86 p^{14} T^{30} + p^{16} T^{32} \)
13 \( 1 - 3 T + 53 T^{2} - 150 T^{3} + 1317 T^{4} - 4452 T^{5} + 23056 T^{6} - 90522 T^{7} + 327950 T^{8} - 1218621 T^{9} + 4082037 T^{10} - 12551466 T^{11} + 42977824 T^{12} - 136810599 T^{13} + 317458436 T^{14} - 1828010007 T^{15} + 2359677564 T^{16} - 1828010007 p T^{17} + 317458436 p^{2} T^{18} - 136810599 p^{3} T^{19} + 42977824 p^{4} T^{20} - 12551466 p^{5} T^{21} + 4082037 p^{6} T^{22} - 1218621 p^{7} T^{23} + 327950 p^{8} T^{24} - 90522 p^{9} T^{25} + 23056 p^{10} T^{26} - 4452 p^{11} T^{27} + 1317 p^{12} T^{28} - 150 p^{13} T^{29} + 53 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \)
17 \( 1 + 9 T - 31 T^{2} - 498 T^{3} + 6 T^{4} + 11613 T^{5} + 1180 T^{6} - 197118 T^{7} + 356723 T^{8} + 3738756 T^{9} - 13114257 T^{10} - 60977685 T^{11} + 321880693 T^{12} + 548795961 T^{13} - 7936687522 T^{14} - 1772823582 T^{15} + 160937528238 T^{16} - 1772823582 p T^{17} - 7936687522 p^{2} T^{18} + 548795961 p^{3} T^{19} + 321880693 p^{4} T^{20} - 60977685 p^{5} T^{21} - 13114257 p^{6} T^{22} + 3738756 p^{7} T^{23} + 356723 p^{8} T^{24} - 197118 p^{9} T^{25} + 1180 p^{10} T^{26} + 11613 p^{11} T^{27} + 6 p^{12} T^{28} - 498 p^{13} T^{29} - 31 p^{14} T^{30} + 9 p^{15} T^{31} + p^{16} T^{32} \)
19 \( 1 + 59 T^{2} + 78 p T^{4} + 279 T^{5} + 21358 T^{6} - 23166 T^{7} + 284075 T^{8} - 1924515 T^{9} + 8259087 T^{10} - 60795468 T^{11} + 210571285 T^{12} - 1125692802 T^{13} + 3051652292 T^{14} - 14070928500 T^{15} + 40349090466 T^{16} - 14070928500 p T^{17} + 3051652292 p^{2} T^{18} - 1125692802 p^{3} T^{19} + 210571285 p^{4} T^{20} - 60795468 p^{5} T^{21} + 8259087 p^{6} T^{22} - 1924515 p^{7} T^{23} + 284075 p^{8} T^{24} - 23166 p^{9} T^{25} + 21358 p^{10} T^{26} + 279 p^{11} T^{27} + 78 p^{13} T^{28} + 59 p^{14} T^{30} + p^{16} T^{32} \)
23 \( 1 - 125 T^{2} + 7680 T^{4} - 295747 T^{6} + 8228447 T^{8} - 202400565 T^{10} + 5673034564 T^{12} - 7469612704 p T^{14} + 4463691201090 T^{16} - 7469612704 p^{3} T^{18} + 5673034564 p^{4} T^{20} - 202400565 p^{6} T^{22} + 8228447 p^{8} T^{24} - 295747 p^{10} T^{26} + 7680 p^{12} T^{28} - 125 p^{14} T^{30} + p^{16} T^{32} \)
29 \( 1 - 6 T + 106 T^{2} - 564 T^{3} + 5445 T^{4} - 23700 T^{5} + 153455 T^{6} - 499140 T^{7} + 2268674 T^{8} - 3839766 T^{9} + 25367247 T^{10} - 108756381 T^{11} + 2283543946 T^{12} - 15452150286 T^{13} + 152754700246 T^{14} - 806878620189 T^{15} + 5726182499856 T^{16} - 806878620189 p T^{17} + 152754700246 p^{2} T^{18} - 15452150286 p^{3} T^{19} + 2283543946 p^{4} T^{20} - 108756381 p^{5} T^{21} + 25367247 p^{6} T^{22} - 3839766 p^{7} T^{23} + 2268674 p^{8} T^{24} - 499140 p^{9} T^{25} + 153455 p^{10} T^{26} - 23700 p^{11} T^{27} + 5445 p^{12} T^{28} - 564 p^{13} T^{29} + 106 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \)
31 \( 1 - 6 T + 128 T^{2} - 696 T^{3} + 7467 T^{4} - 41745 T^{5} + 298711 T^{6} - 1778985 T^{7} + 10180130 T^{8} - 58737741 T^{9} + 320070333 T^{10} - 1687070526 T^{11} + 8941505953 T^{12} - 50372743224 T^{13} + 219778671713 T^{14} - 1637243989362 T^{15} + 5894911299924 T^{16} - 1637243989362 p T^{17} + 219778671713 p^{2} T^{18} - 50372743224 p^{3} T^{19} + 8941505953 p^{4} T^{20} - 1687070526 p^{5} T^{21} + 320070333 p^{6} T^{22} - 58737741 p^{7} T^{23} + 10180130 p^{8} T^{24} - 1778985 p^{9} T^{25} + 298711 p^{10} T^{26} - 41745 p^{11} T^{27} + 7467 p^{12} T^{28} - 696 p^{13} T^{29} + 128 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \)
37 \( 1 - T - 188 T^{2} + 55 T^{3} + 18431 T^{4} + 6958 T^{5} - 1220598 T^{6} - 1466820 T^{7} + 61023285 T^{8} + 128763363 T^{9} - 65590950 p T^{10} - 7087005807 T^{11} + 80710069782 T^{12} + 247085609469 T^{13} - 2444398029834 T^{14} - 3801878390997 T^{15} + 81128049964254 T^{16} - 3801878390997 p T^{17} - 2444398029834 p^{2} T^{18} + 247085609469 p^{3} T^{19} + 80710069782 p^{4} T^{20} - 7087005807 p^{5} T^{21} - 65590950 p^{7} T^{22} + 128763363 p^{7} T^{23} + 61023285 p^{8} T^{24} - 1466820 p^{9} T^{25} - 1220598 p^{10} T^{26} + 6958 p^{11} T^{27} + 18431 p^{12} T^{28} + 55 p^{13} T^{29} - 188 p^{14} T^{30} - p^{15} T^{31} + p^{16} T^{32} \)
41 \( 1 - 6 T - 142 T^{2} + 1146 T^{3} + 7575 T^{4} - 89148 T^{5} - 186515 T^{6} + 3858942 T^{7} + 5165546 T^{8} - 137269398 T^{9} - 383379603 T^{10} + 6275799867 T^{11} + 14048576428 T^{12} - 262470296220 T^{13} + 63020850350 T^{14} + 4818424458177 T^{15} - 18114488698896 T^{16} + 4818424458177 p T^{17} + 63020850350 p^{2} T^{18} - 262470296220 p^{3} T^{19} + 14048576428 p^{4} T^{20} + 6275799867 p^{5} T^{21} - 383379603 p^{6} T^{22} - 137269398 p^{7} T^{23} + 5165546 p^{8} T^{24} + 3858942 p^{9} T^{25} - 186515 p^{10} T^{26} - 89148 p^{11} T^{27} + 7575 p^{12} T^{28} + 1146 p^{13} T^{29} - 142 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \)
43 \( 1 + 2 T - 137 T^{2} - 620 T^{3} + 8660 T^{4} + 62377 T^{5} - 131688 T^{6} - 3324402 T^{7} - 16902063 T^{8} + 47132751 T^{9} + 1398120027 T^{10} + 5063156292 T^{11} - 36164979717 T^{12} - 388120390392 T^{13} - 815732360646 T^{14} + 8046953157708 T^{15} + 93229603031718 T^{16} + 8046953157708 p T^{17} - 815732360646 p^{2} T^{18} - 388120390392 p^{3} T^{19} - 36164979717 p^{4} T^{20} + 5063156292 p^{5} T^{21} + 1398120027 p^{6} T^{22} + 47132751 p^{7} T^{23} - 16902063 p^{8} T^{24} - 3324402 p^{9} T^{25} - 131688 p^{10} T^{26} + 62377 p^{11} T^{27} + 8660 p^{12} T^{28} - 620 p^{13} T^{29} - 137 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \)
47 \( 1 + 18 T + 62 T^{2} - 192 T^{3} + 3579 T^{4} + 45093 T^{5} + 305398 T^{6} + 2890521 T^{7} + 680339 T^{8} - 61965303 T^{9} + 840301383 T^{10} + 4943829948 T^{11} + 3397740844 T^{12} + 111292595154 T^{13} - 711051013816 T^{14} - 459408198168 T^{15} + 101830826927016 T^{16} - 459408198168 p T^{17} - 711051013816 p^{2} T^{18} + 111292595154 p^{3} T^{19} + 3397740844 p^{4} T^{20} + 4943829948 p^{5} T^{21} + 840301383 p^{6} T^{22} - 61965303 p^{7} T^{23} + 680339 p^{8} T^{24} + 2890521 p^{9} T^{25} + 305398 p^{10} T^{26} + 45093 p^{11} T^{27} + 3579 p^{12} T^{28} - 192 p^{13} T^{29} + 62 p^{14} T^{30} + 18 p^{15} T^{31} + p^{16} T^{32} \)
53 \( 1 + 271 T^{2} + 39798 T^{4} - 8019 T^{5} + 3882470 T^{6} - 3112830 T^{7} + 272618249 T^{8} - 573793713 T^{9} + 13751201223 T^{10} - 69938411256 T^{11} + 461723008303 T^{12} - 6218775316710 T^{13} + 7202514917530 T^{14} - 423085276358208 T^{15} - 24546838659498 T^{16} - 423085276358208 p T^{17} + 7202514917530 p^{2} T^{18} - 6218775316710 p^{3} T^{19} + 461723008303 p^{4} T^{20} - 69938411256 p^{5} T^{21} + 13751201223 p^{6} T^{22} - 573793713 p^{7} T^{23} + 272618249 p^{8} T^{24} - 3112830 p^{9} T^{25} + 3882470 p^{10} T^{26} - 8019 p^{11} T^{27} + 39798 p^{12} T^{28} + 271 p^{14} T^{30} + p^{16} T^{32} \)
59 \( 1 + 15 T - 79 T^{2} - 1428 T^{3} + 12489 T^{4} + 104847 T^{5} - 1041029 T^{6} + 349611 T^{7} + 90472109 T^{8} - 405874929 T^{9} - 2700289995 T^{10} + 51789934650 T^{11} - 105592258130 T^{12} - 2611128991413 T^{13} + 30211727298980 T^{14} + 82730812140738 T^{15} - 2120569425031284 T^{16} + 82730812140738 p T^{17} + 30211727298980 p^{2} T^{18} - 2611128991413 p^{3} T^{19} - 105592258130 p^{4} T^{20} + 51789934650 p^{5} T^{21} - 2700289995 p^{6} T^{22} - 405874929 p^{7} T^{23} + 90472109 p^{8} T^{24} + 349611 p^{9} T^{25} - 1041029 p^{10} T^{26} + 104847 p^{11} T^{27} + 12489 p^{12} T^{28} - 1428 p^{13} T^{29} - 79 p^{14} T^{30} + 15 p^{15} T^{31} + p^{16} T^{32} \)
61 \( 1 - 3 T + 377 T^{2} - 1122 T^{3} + 1221 p T^{4} - 239253 T^{5} + 10537021 T^{6} - 36111681 T^{7} + 1196449751 T^{8} - 4163017557 T^{9} + 114788648541 T^{10} - 392960339076 T^{11} + 9549397216930 T^{12} - 31727661059931 T^{13} + 697503406538990 T^{14} - 2222103948983802 T^{15} + 45117478421428590 T^{16} - 2222103948983802 p T^{17} + 697503406538990 p^{2} T^{18} - 31727661059931 p^{3} T^{19} + 9549397216930 p^{4} T^{20} - 392960339076 p^{5} T^{21} + 114788648541 p^{6} T^{22} - 4163017557 p^{7} T^{23} + 1196449751 p^{8} T^{24} - 36111681 p^{9} T^{25} + 10537021 p^{10} T^{26} - 239253 p^{11} T^{27} + 1221 p^{13} T^{28} - 1122 p^{13} T^{29} + 377 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \)
67 \( 1 + 7 T - 266 T^{2} - 955 T^{3} + 40565 T^{4} + 13739 T^{5} - 4302801 T^{6} + 7895916 T^{7} + 333320832 T^{8} - 1176952818 T^{9} - 19419549429 T^{10} + 109762768995 T^{11} + 793012008891 T^{12} - 6957819527658 T^{13} - 12193565486865 T^{14} + 198908265810519 T^{15} - 461225363558112 T^{16} + 198908265810519 p T^{17} - 12193565486865 p^{2} T^{18} - 6957819527658 p^{3} T^{19} + 793012008891 p^{4} T^{20} + 109762768995 p^{5} T^{21} - 19419549429 p^{6} T^{22} - 1176952818 p^{7} T^{23} + 333320832 p^{8} T^{24} + 7895916 p^{9} T^{25} - 4302801 p^{10} T^{26} + 13739 p^{11} T^{27} + 40565 p^{12} T^{28} - 955 p^{13} T^{29} - 266 p^{14} T^{30} + 7 p^{15} T^{31} + p^{16} T^{32} \)
71 \( 1 - 389 T^{2} + 95718 T^{4} - 239186 p T^{6} + 2417461733 T^{8} - 285171142929 T^{10} + 28739565041956 T^{12} - 2501948752776044 T^{14} + 190114711967546784 T^{16} - 2501948752776044 p^{2} T^{18} + 28739565041956 p^{4} T^{20} - 285171142929 p^{6} T^{22} + 2417461733 p^{8} T^{24} - 239186 p^{11} T^{26} + 95718 p^{12} T^{28} - 389 p^{14} T^{30} + p^{16} T^{32} \)
73 \( 1 + 278 T^{2} + 39699 T^{4} - 160362 T^{5} + 4049527 T^{6} - 43185312 T^{7} + 316493258 T^{8} - 5978310966 T^{9} + 30516141699 T^{10} - 573490190745 T^{11} + 3892519182646 T^{12} - 40210747978752 T^{13} + 436575836052170 T^{14} - 2488085380264887 T^{15} + 37348173312201540 T^{16} - 2488085380264887 p T^{17} + 436575836052170 p^{2} T^{18} - 40210747978752 p^{3} T^{19} + 3892519182646 p^{4} T^{20} - 573490190745 p^{5} T^{21} + 30516141699 p^{6} T^{22} - 5978310966 p^{7} T^{23} + 316493258 p^{8} T^{24} - 43185312 p^{9} T^{25} + 4049527 p^{10} T^{26} - 160362 p^{11} T^{27} + 39699 p^{12} T^{28} + 278 p^{14} T^{30} + p^{16} T^{32} \)
79 \( 1 + T - 488 T^{2} + 305 T^{3} + 130067 T^{4} - 217447 T^{5} - 23515767 T^{6} + 58439790 T^{7} + 3200266602 T^{8} - 9493225932 T^{9} - 346875510927 T^{10} + 1047490944351 T^{11} + 31673904979161 T^{12} - 75618180243888 T^{13} - 2595852946891617 T^{14} + 2451851231878899 T^{15} + 205361771983476300 T^{16} + 2451851231878899 p T^{17} - 2595852946891617 p^{2} T^{18} - 75618180243888 p^{3} T^{19} + 31673904979161 p^{4} T^{20} + 1047490944351 p^{5} T^{21} - 346875510927 p^{6} T^{22} - 9493225932 p^{7} T^{23} + 3200266602 p^{8} T^{24} + 58439790 p^{9} T^{25} - 23515767 p^{10} T^{26} - 217447 p^{11} T^{27} + 130067 p^{12} T^{28} + 305 p^{13} T^{29} - 488 p^{14} T^{30} + p^{15} T^{31} + p^{16} T^{32} \)
83 \( 1 - 334 T^{2} + 1272 T^{3} + 54816 T^{4} - 395796 T^{5} - 5397056 T^{6} + 60133572 T^{7} + 310945691 T^{8} - 6455120304 T^{9} - 3389068248 T^{10} + 563598757704 T^{11} - 2368219763012 T^{12} - 39202056035964 T^{13} + 437295746267318 T^{14} + 1352773893773628 T^{15} - 45259898774669136 T^{16} + 1352773893773628 p T^{17} + 437295746267318 p^{2} T^{18} - 39202056035964 p^{3} T^{19} - 2368219763012 p^{4} T^{20} + 563598757704 p^{5} T^{21} - 3389068248 p^{6} T^{22} - 6455120304 p^{7} T^{23} + 310945691 p^{8} T^{24} + 60133572 p^{9} T^{25} - 5397056 p^{10} T^{26} - 395796 p^{11} T^{27} + 54816 p^{12} T^{28} + 1272 p^{13} T^{29} - 334 p^{14} T^{30} + p^{16} T^{32} \)
89 \( 1 + 21 T - 253 T^{2} - 6084 T^{3} + 64134 T^{4} + 1086543 T^{5} - 14444834 T^{6} - 154075794 T^{7} + 2457441293 T^{8} + 16916386392 T^{9} - 343956627405 T^{10} - 1437006555873 T^{11} + 41653185355711 T^{12} + 92739461592729 T^{13} - 4399434577615834 T^{14} - 3097874885577318 T^{15} + 411982470004053510 T^{16} - 3097874885577318 p T^{17} - 4399434577615834 p^{2} T^{18} + 92739461592729 p^{3} T^{19} + 41653185355711 p^{4} T^{20} - 1437006555873 p^{5} T^{21} - 343956627405 p^{6} T^{22} + 16916386392 p^{7} T^{23} + 2457441293 p^{8} T^{24} - 154075794 p^{9} T^{25} - 14444834 p^{10} T^{26} + 1086543 p^{11} T^{27} + 64134 p^{12} T^{28} - 6084 p^{13} T^{29} - 253 p^{14} T^{30} + 21 p^{15} T^{31} + p^{16} T^{32} \)
97 \( 1 - 3 T + 392 T^{2} - 1167 T^{3} + 75243 T^{4} - 248802 T^{5} + 9281977 T^{6} - 31870980 T^{7} + 787182476 T^{8} - 2086019346 T^{9} + 40966656945 T^{10} + 99548858823 T^{11} - 1131878124686 T^{12} + 48785193558327 T^{13} - 618766130152816 T^{14} + 7413615555033420 T^{15} - 81803373889459032 T^{16} + 7413615555033420 p T^{17} - 618766130152816 p^{2} T^{18} + 48785193558327 p^{3} T^{19} - 1131878124686 p^{4} T^{20} + 99548858823 p^{5} T^{21} + 40966656945 p^{6} T^{22} - 2086019346 p^{7} T^{23} + 787182476 p^{8} T^{24} - 31870980 p^{9} T^{25} + 9281977 p^{10} T^{26} - 248802 p^{11} T^{27} + 75243 p^{12} T^{28} - 1167 p^{13} T^{29} + 392 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.59185170081974633967349128685, −3.40869552281640422237334656336, −3.34298994217974885809136041587, −3.24347786379755255567568035890, −3.05368711137577362228264469935, −2.94311735877362145157990498035, −2.91342194540165627773136029488, −2.71373001421804872193086746886, −2.69236532658584826482278878805, −2.64944811537694575535911338650, −2.52941968447915270475110343846, −2.33043968662820976675585624333, −2.14947483260548795292350632890, −2.09066601938496500185619370109, −2.03184608511635421057746453389, −1.97378115836210442940779006582, −1.73604667345096736976918084728, −1.72115399198710913501927001541, −1.70651529942427102631710525937, −1.53847253779047415068641405102, −1.26248103100685459051445891718, −1.05252846935472732246611356843, −0.68510219330231687961062531812, −0.42177171080692285946820330742, −0.38968082043150080744755969090, 0.38968082043150080744755969090, 0.42177171080692285946820330742, 0.68510219330231687961062531812, 1.05252846935472732246611356843, 1.26248103100685459051445891718, 1.53847253779047415068641405102, 1.70651529942427102631710525937, 1.72115399198710913501927001541, 1.73604667345096736976918084728, 1.97378115836210442940779006582, 2.03184608511635421057746453389, 2.09066601938496500185619370109, 2.14947483260548795292350632890, 2.33043968662820976675585624333, 2.52941968447915270475110343846, 2.64944811537694575535911338650, 2.69236532658584826482278878805, 2.71373001421804872193086746886, 2.91342194540165627773136029488, 2.94311735877362145157990498035, 3.05368711137577362228264469935, 3.24347786379755255567568035890, 3.34298994217974885809136041587, 3.40869552281640422237334656336, 3.59185170081974633967349128685

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.