Properties

Label 32-42e32-1.1-c1e16-0-6
Degree $32$
Conductor $8.790\times 10^{51}$
Sign $1$
Analytic cond. $2.40110\times 10^{18}$
Root an. cond. $3.75308$
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  − 3·9-s + 6·11-s + 3·13-s + 18·17-s − 21·23-s + 16·25-s − 3·27-s + 6·29-s − 6·31-s − 2·37-s + 6·41-s − 2·43-s − 18·47-s − 15·59-s − 3·61-s − 7·67-s − 79-s + 42·89-s + 3·97-s − 18·99-s − 24·101-s − 21·103-s + 28·109-s − 9·113-s − 9·117-s − 25·121-s − 24·125-s + ⋯
L(s)  = 1  − 9-s + 1.80·11-s + 0.832·13-s + 4.36·17-s − 4.37·23-s + 16/5·25-s − 0.577·27-s + 1.11·29-s − 1.07·31-s − 0.328·37-s + 0.937·41-s − 0.304·43-s − 2.62·47-s − 1.95·59-s − 0.384·61-s − 0.855·67-s − 0.112·79-s + 4.45·89-s + 0.304·97-s − 1.80·99-s − 2.38·101-s − 2.06·103-s + 2.68·109-s − 0.846·113-s − 0.832·117-s − 2.27·121-s − 2.14·125-s + ⋯

Functional equation

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

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{32} \cdot 7^{32}\)
Sign: $1$
Analytic conductor: \(2.40110\times 10^{18}\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.810644544\)
\(L(\frac12)\) \(\approx\) \(7.810644544\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} + p T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + 5 p^{2} T^{6} + p^{2} T^{7} + 19 p^{2} T^{8} + p^{3} T^{9} + 5 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} + p^{6} T^{13} + p^{7} T^{14} + p^{8} T^{16} \)
7 \( 1 \)
good5 \( 1 - 16 T^{2} + 24 T^{3} + 21 p T^{4} - 357 T^{5} - 16 p T^{6} + 2232 T^{7} - 2857 T^{8} - 6027 T^{9} + 16041 T^{10} - 777 T^{11} - 31496 T^{12} + 32724 T^{13} + 43664 T^{14} - 15327 T^{15} - 229716 T^{16} - 15327 p T^{17} + 43664 p^{2} T^{18} + 32724 p^{3} T^{19} - 31496 p^{4} T^{20} - 777 p^{5} T^{21} + 16041 p^{6} T^{22} - 6027 p^{7} T^{23} - 2857 p^{8} T^{24} + 2232 p^{9} T^{25} - 16 p^{11} T^{26} - 357 p^{11} T^{27} + 21 p^{13} T^{28} + 24 p^{13} T^{29} - 16 p^{14} T^{30} + p^{16} T^{32} \)
11 \( 1 - 6 T + 61 T^{2} - 294 T^{3} + 1701 T^{4} - 564 p T^{5} + 24845 T^{6} - 59553 T^{7} + 143666 T^{8} + 78393 T^{9} - 1097631 T^{10} + 8096325 T^{11} - 21925205 T^{12} + 68059401 T^{13} - 34454504 T^{14} - 43343730 T^{15} + 1352446632 T^{16} - 43343730 p T^{17} - 34454504 p^{2} T^{18} + 68059401 p^{3} T^{19} - 21925205 p^{4} T^{20} + 8096325 p^{5} T^{21} - 1097631 p^{6} T^{22} + 78393 p^{7} T^{23} + 143666 p^{8} T^{24} - 59553 p^{9} T^{25} + 24845 p^{10} T^{26} - 564 p^{12} T^{27} + 1701 p^{12} T^{28} - 294 p^{13} T^{29} + 61 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \)
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 \( ( 1 - 9 T + 112 T^{2} - 753 T^{3} + 5761 T^{4} - 30975 T^{5} + 179089 T^{6} - 791448 T^{7} + 3696319 T^{8} - 791448 p T^{9} + 179089 p^{2} T^{10} - 30975 p^{3} T^{11} + 5761 p^{4} T^{12} - 753 p^{5} T^{13} + 112 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
19 \( 1 - 118 T^{2} + 7479 T^{4} - 332465 T^{6} + 11419853 T^{8} - 321845868 T^{10} + 7780360357 T^{12} - 167488957369 T^{14} + 3305457417861 T^{16} - 167488957369 p^{2} T^{18} + 7780360357 p^{4} T^{20} - 321845868 p^{6} T^{22} + 11419853 p^{8} T^{24} - 332465 p^{10} T^{26} + 7479 p^{12} T^{28} - 118 p^{14} T^{30} + p^{16} T^{32} \)
23 \( 1 + 21 T + 283 T^{2} + 2856 T^{3} + 23904 T^{4} + 177009 T^{5} + 1214978 T^{6} + 7994925 T^{7} + 50784536 T^{8} + 310700127 T^{9} + 1817756595 T^{10} + 10184685783 T^{11} + 55106115085 T^{12} + 290568927984 T^{13} + 1499561670325 T^{14} + 7546165977213 T^{15} + 36821082385998 T^{16} + 7546165977213 p T^{17} + 1499561670325 p^{2} T^{18} + 290568927984 p^{3} T^{19} + 55106115085 p^{4} T^{20} + 10184685783 p^{5} T^{21} + 1817756595 p^{6} T^{22} + 310700127 p^{7} T^{23} + 50784536 p^{8} T^{24} + 7994925 p^{9} T^{25} + 1214978 p^{10} T^{26} + 177009 p^{11} T^{27} + 23904 p^{12} T^{28} + 2856 p^{13} T^{29} + 283 p^{14} T^{30} + 21 p^{15} T^{31} + p^{16} T^{32} \)
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 \( ( 1 + T + 189 T^{2} + 122 T^{3} + 464 p T^{4} + 4320 T^{5} + 1006795 T^{6} - 1511 T^{7} + 42902622 T^{8} - 1511 p T^{9} + 1006795 p^{2} T^{10} + 4320 p^{3} T^{11} + 464 p^{5} T^{12} + 122 p^{5} T^{13} + 189 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} )^{2} \)
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 \( 1 - 542 T^{2} + 140727 T^{4} - 23469289 T^{6} + 2850317849 T^{8} - 5117135592 p T^{10} + 21201779873365 T^{12} - 1403502501960773 T^{14} + 79972351708005549 T^{16} - 1403502501960773 p^{2} T^{18} + 21201779873365 p^{4} T^{20} - 5117135592 p^{7} T^{22} + 2850317849 p^{8} T^{24} - 23469289 p^{10} T^{26} + 140727 p^{12} T^{28} - 542 p^{14} T^{30} + p^{16} T^{32} \)
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 \( 1 - 556 T^{2} + 152454 T^{4} - 27296363 T^{6} + 3604910159 T^{8} - 380031280554 T^{10} + 34108990053451 T^{12} - 2750292903642481 T^{14} + 206651113497718443 T^{16} - 2750292903642481 p^{2} T^{18} + 34108990053451 p^{4} T^{20} - 380031280554 p^{6} T^{22} + 3604910159 p^{8} T^{24} - 27296363 p^{10} T^{26} + 152454 p^{12} T^{28} - 556 p^{14} T^{30} + p^{16} T^{32} \)
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 \( ( 1 - 21 T + 694 T^{2} - 10329 T^{3} + 200593 T^{4} - 2343123 T^{5} + 33642025 T^{6} - 319771308 T^{7} + 3660822475 T^{8} - 319771308 p T^{9} + 33642025 p^{2} T^{10} - 2343123 p^{3} T^{11} + 200593 p^{4} T^{12} - 10329 p^{5} T^{13} + 694 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
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

−2.27598053506166116441107281984, −2.22722916296479221725113774630, −2.20870753565220419705889010343, −2.18983849762186635463974893560, −2.18800663183378830740874382766, −1.88176204295001027192528452724, −1.78796362771335283523216435419, −1.75334161407476909997222436513, −1.56299645584057732090097259789, −1.56008832104693517386352056374, −1.51593318885007786136303585874, −1.50657208950268805323975477257, −1.36320412147329382305526678779, −1.32989334774304490716643011722, −1.30119412084964094718634118278, −1.13699843894073729345121189893, −1.11857787356516582090747262251, −0.984960233914116200443141338435, −0.889098812284665070371060728257, −0.854027405594720923317485600640, −0.57864151116930339058354825044, −0.40197861182711005146416132304, −0.32809519256753680341598759058, −0.24818145278534970541957540140, −0.14294206158507803587721277543, 0.14294206158507803587721277543, 0.24818145278534970541957540140, 0.32809519256753680341598759058, 0.40197861182711005146416132304, 0.57864151116930339058354825044, 0.854027405594720923317485600640, 0.889098812284665070371060728257, 0.984960233914116200443141338435, 1.11857787356516582090747262251, 1.13699843894073729345121189893, 1.30119412084964094718634118278, 1.32989334774304490716643011722, 1.36320412147329382305526678779, 1.50657208950268805323975477257, 1.51593318885007786136303585874, 1.56008832104693517386352056374, 1.56299645584057732090097259789, 1.75334161407476909997222436513, 1.78796362771335283523216435419, 1.88176204295001027192528452724, 2.18800663183378830740874382766, 2.18983849762186635463974893560, 2.20870753565220419705889010343, 2.22722916296479221725113774630, 2.27598053506166116441107281984

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

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