Properties

Label 32-1040e16-1.1-c1e16-0-0
Degree $32$
Conductor $1.873\times 10^{48}$
Sign $1$
Analytic cond. $5.11643\times 10^{14}$
Root an. cond. $2.88174$
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  + 4·3-s − 6·7-s + 12·9-s + 6·11-s − 2·13-s + 4·17-s − 30·19-s − 24·21-s − 6·23-s − 8·25-s + 20·27-s − 16·29-s + 24·33-s − 24·37-s − 8·39-s − 24·41-s − 6·43-s − 12·49-s + 16·51-s + 4·53-s − 120·57-s − 12·59-s − 2·61-s − 72·63-s + 6·67-s − 24·69-s − 72·71-s + ⋯
L(s)  = 1  + 2.30·3-s − 2.26·7-s + 4·9-s + 1.80·11-s − 0.554·13-s + 0.970·17-s − 6.88·19-s − 5.23·21-s − 1.25·23-s − 8/5·25-s + 3.84·27-s − 2.97·29-s + 4.17·33-s − 3.94·37-s − 1.28·39-s − 3.74·41-s − 0.914·43-s − 1.71·49-s + 2.24·51-s + 0.549·53-s − 15.8·57-s − 1.56·59-s − 0.256·61-s − 9.07·63-s + 0.733·67-s − 2.88·69-s − 8.54·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{64} \cdot 5^{16} \cdot 13^{16}\)
Sign: $1$
Analytic conductor: \(5.11643\times 10^{14}\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{64} \cdot 5^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.01588020997\)
\(L(\frac12)\) \(\approx\) \(0.01588020997\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 + T^{2} )^{8} \)
13 \( 1 + 2 T + 38 T^{2} + 68 T^{3} + 985 T^{4} + 1928 T^{5} + 18874 T^{6} + 2514 p T^{7} + 273420 T^{8} + 2514 p^{2} T^{9} + 18874 p^{2} T^{10} + 1928 p^{3} T^{11} + 985 p^{4} T^{12} + 68 p^{5} T^{13} + 38 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \)
good3 \( 1 - 4 T + 4 T^{2} + 4 p T^{3} - 5 p^{2} T^{4} + 56 T^{5} + 20 p T^{6} - 40 p^{2} T^{7} + 517 T^{8} + 68 T^{9} - 1424 T^{10} + 2524 T^{11} - 490 p T^{12} - 92 p^{3} T^{13} + 5624 T^{14} - 3628 T^{15} + 1402 T^{16} - 3628 p T^{17} + 5624 p^{2} T^{18} - 92 p^{6} T^{19} - 490 p^{5} T^{20} + 2524 p^{5} T^{21} - 1424 p^{6} T^{22} + 68 p^{7} T^{23} + 517 p^{8} T^{24} - 40 p^{11} T^{25} + 20 p^{11} T^{26} + 56 p^{11} T^{27} - 5 p^{14} T^{28} + 4 p^{14} T^{29} + 4 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \)
7 \( 1 + 6 T + 48 T^{2} + 216 T^{3} + 1062 T^{4} + 4194 T^{5} + 16332 T^{6} + 8418 p T^{7} + 198697 T^{8} + 658482 T^{9} + 2027352 T^{10} + 6219954 T^{11} + 2585682 p T^{12} + 7417800 p T^{13} + 144940524 T^{14} + 56182278 p T^{15} + 1057744164 T^{16} + 56182278 p^{2} T^{17} + 144940524 p^{2} T^{18} + 7417800 p^{4} T^{19} + 2585682 p^{5} T^{20} + 6219954 p^{5} T^{21} + 2027352 p^{6} T^{22} + 658482 p^{7} T^{23} + 198697 p^{8} T^{24} + 8418 p^{10} T^{25} + 16332 p^{10} T^{26} + 4194 p^{11} T^{27} + 1062 p^{12} T^{28} + 216 p^{13} T^{29} + 48 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \)
11 \( 1 - 6 T + 24 T^{2} - 72 T^{3} + 2 p^{2} T^{4} - 1026 T^{5} + 2588 T^{6} - 186 p T^{7} - 16915 T^{8} + 83118 T^{9} - 261584 T^{10} + 527214 T^{11} - 186970 p T^{12} + 975552 p T^{13} - 28172196 T^{14} + 1032786 p T^{15} + 238188124 T^{16} + 1032786 p^{2} T^{17} - 28172196 p^{2} T^{18} + 975552 p^{4} T^{19} - 186970 p^{5} T^{20} + 527214 p^{5} T^{21} - 261584 p^{6} T^{22} + 83118 p^{7} T^{23} - 16915 p^{8} T^{24} - 186 p^{10} T^{25} + 2588 p^{10} T^{26} - 1026 p^{11} T^{27} + 2 p^{14} T^{28} - 72 p^{13} T^{29} + 24 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \)
17 \( 1 - 4 T - 32 T^{2} - 12 T^{3} + 923 T^{4} + 3460 T^{5} - 4280 T^{6} - 133036 T^{7} - 248091 T^{8} + 1589320 T^{9} + 8955416 T^{10} + 5315840 T^{11} - 95059470 T^{12} - 580633904 T^{13} + 67673152 T^{14} + 4374099136 T^{15} + 21633136170 T^{16} + 4374099136 p T^{17} + 67673152 p^{2} T^{18} - 580633904 p^{3} T^{19} - 95059470 p^{4} T^{20} + 5315840 p^{5} T^{21} + 8955416 p^{6} T^{22} + 1589320 p^{7} T^{23} - 248091 p^{8} T^{24} - 133036 p^{9} T^{25} - 4280 p^{10} T^{26} + 3460 p^{11} T^{27} + 923 p^{12} T^{28} - 12 p^{13} T^{29} - 32 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \)
19 \( 1 + 30 T + 540 T^{2} + 7200 T^{3} + 78358 T^{4} + 727626 T^{5} + 5937732 T^{6} + 43395798 T^{7} + 288194673 T^{8} + 1758682098 T^{9} + 9959215008 T^{10} + 52821338658 T^{11} + 264851164238 T^{12} + 1267818674448 T^{13} + 5851651168224 T^{14} + 26281091231706 T^{15} + 115608523761524 T^{16} + 26281091231706 p T^{17} + 5851651168224 p^{2} T^{18} + 1267818674448 p^{3} T^{19} + 264851164238 p^{4} T^{20} + 52821338658 p^{5} T^{21} + 9959215008 p^{6} T^{22} + 1758682098 p^{7} T^{23} + 288194673 p^{8} T^{24} + 43395798 p^{9} T^{25} + 5937732 p^{10} T^{26} + 727626 p^{11} T^{27} + 78358 p^{12} T^{28} + 7200 p^{13} T^{29} + 540 p^{14} T^{30} + 30 p^{15} T^{31} + p^{16} T^{32} \)
23 \( 1 + 6 T - 74 T^{2} - 564 T^{3} + 1939 T^{4} + 21000 T^{5} - 27058 T^{6} - 412002 T^{7} + 1032113 T^{8} + 6119604 T^{9} - 53474360 T^{10} - 5370996 p T^{11} + 1772145230 T^{12} + 2328566568 T^{13} - 47944093876 T^{14} - 19498999044 T^{15} + 1175990856538 T^{16} - 19498999044 p T^{17} - 47944093876 p^{2} T^{18} + 2328566568 p^{3} T^{19} + 1772145230 p^{4} T^{20} - 5370996 p^{6} T^{21} - 53474360 p^{6} T^{22} + 6119604 p^{7} T^{23} + 1032113 p^{8} T^{24} - 412002 p^{9} T^{25} - 27058 p^{10} T^{26} + 21000 p^{11} T^{27} + 1939 p^{12} T^{28} - 564 p^{13} T^{29} - 74 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \)
29 \( 1 + 16 T - 30 T^{2} - 1632 T^{3} + 187 T^{4} + 125824 T^{5} + 195326 T^{6} - 6523216 T^{7} - 16126567 T^{8} + 272013152 T^{9} + 971794948 T^{10} - 8540874464 T^{11} - 43702361762 T^{12} + 196735147776 T^{13} + 1641311048792 T^{14} - 2158192906016 T^{15} - 51561131514286 T^{16} - 2158192906016 p T^{17} + 1641311048792 p^{2} T^{18} + 196735147776 p^{3} T^{19} - 43702361762 p^{4} T^{20} - 8540874464 p^{5} T^{21} + 971794948 p^{6} T^{22} + 272013152 p^{7} T^{23} - 16126567 p^{8} T^{24} - 6523216 p^{9} T^{25} + 195326 p^{10} T^{26} + 125824 p^{11} T^{27} + 187 p^{12} T^{28} - 1632 p^{13} T^{29} - 30 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \)
31 \( 1 - 208 T^{2} + 23444 T^{4} - 1882464 T^{6} + 119109732 T^{8} - 6223272576 T^{10} + 275855496748 T^{12} - 10541276996432 T^{14} + 350110697164214 T^{16} - 10541276996432 p^{2} T^{18} + 275855496748 p^{4} T^{20} - 6223272576 p^{6} T^{22} + 119109732 p^{8} T^{24} - 1882464 p^{10} T^{26} + 23444 p^{12} T^{28} - 208 p^{14} T^{30} + p^{16} T^{32} \)
37 \( 1 + 24 T + 460 T^{2} + 6432 T^{3} + 78858 T^{4} + 22440 p T^{5} + 7877848 T^{6} + 67168152 T^{7} + 522615053 T^{8} + 3711800136 T^{9} + 24117143168 T^{10} + 143201691240 T^{11} + 775835553186 T^{12} + 3847164126864 T^{13} + 17834814939644 T^{14} + 82827862708104 T^{15} + 446732143406188 T^{16} + 82827862708104 p T^{17} + 17834814939644 p^{2} T^{18} + 3847164126864 p^{3} T^{19} + 775835553186 p^{4} T^{20} + 143201691240 p^{5} T^{21} + 24117143168 p^{6} T^{22} + 3711800136 p^{7} T^{23} + 522615053 p^{8} T^{24} + 67168152 p^{9} T^{25} + 7877848 p^{10} T^{26} + 22440 p^{12} T^{27} + 78858 p^{12} T^{28} + 6432 p^{13} T^{29} + 460 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \)
41 \( 1 + 24 T + 462 T^{2} + 6480 T^{3} + 78523 T^{4} + 818952 T^{5} + 7680330 T^{6} + 64942800 T^{7} + 504179193 T^{8} + 3587911704 T^{9} + 23682609012 T^{10} + 144169303656 T^{11} + 819524037758 T^{12} + 4353443943768 T^{13} + 22403620517904 T^{14} + 117594304534824 T^{15} + 694944352237634 T^{16} + 117594304534824 p T^{17} + 22403620517904 p^{2} T^{18} + 4353443943768 p^{3} T^{19} + 819524037758 p^{4} T^{20} + 144169303656 p^{5} T^{21} + 23682609012 p^{6} T^{22} + 3587911704 p^{7} T^{23} + 504179193 p^{8} T^{24} + 64942800 p^{9} T^{25} + 7680330 p^{10} T^{26} + 818952 p^{11} T^{27} + 78523 p^{12} T^{28} + 6480 p^{13} T^{29} + 462 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \)
43 \( 1 + 6 T - 190 T^{2} - 748 T^{3} + 19635 T^{4} + 34448 T^{5} - 1540406 T^{6} - 1124418 T^{7} + 99144449 T^{8} + 45488916 T^{9} - 5668105944 T^{10} - 688081276 T^{11} + 300854507870 T^{12} - 25928432304 T^{13} - 14145282211020 T^{14} + 661022192892 T^{15} + 612083434948746 T^{16} + 661022192892 p T^{17} - 14145282211020 p^{2} T^{18} - 25928432304 p^{3} T^{19} + 300854507870 p^{4} T^{20} - 688081276 p^{5} T^{21} - 5668105944 p^{6} T^{22} + 45488916 p^{7} T^{23} + 99144449 p^{8} T^{24} - 1124418 p^{9} T^{25} - 1540406 p^{10} T^{26} + 34448 p^{11} T^{27} + 19635 p^{12} T^{28} - 748 p^{13} T^{29} - 190 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \)
47 \( 1 - 364 T^{2} + 69798 T^{4} - 9185416 T^{6} + 923415089 T^{8} - 75058727912 T^{10} + 5107567284342 T^{12} - 297310111361252 T^{14} + 14986650675462244 T^{16} - 297310111361252 p^{2} T^{18} + 5107567284342 p^{4} T^{20} - 75058727912 p^{6} T^{22} + 923415089 p^{8} T^{24} - 9185416 p^{10} T^{26} + 69798 p^{12} T^{28} - 364 p^{14} T^{30} + p^{16} T^{32} \)
53 \( ( 1 - 2 T + 194 T^{2} - 928 T^{3} + 22445 T^{4} - 118984 T^{5} + 1882854 T^{6} - 9454190 T^{7} + 114806484 T^{8} - 9454190 p T^{9} + 1882854 p^{2} T^{10} - 118984 p^{3} T^{11} + 22445 p^{4} T^{12} - 928 p^{5} T^{13} + 194 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
59 \( 1 + 12 T + 274 T^{2} + 2712 T^{3} + 33659 T^{4} + 267528 T^{5} + 2390350 T^{6} + 14111412 T^{7} + 94591561 T^{8} + 257608992 T^{9} + 146376772 T^{10} - 29057164704 T^{11} - 306298241714 T^{12} - 4041397703880 T^{13} - 31649904259944 T^{14} - 321601760278512 T^{15} - 2184202627142670 T^{16} - 321601760278512 p T^{17} - 31649904259944 p^{2} T^{18} - 4041397703880 p^{3} T^{19} - 306298241714 p^{4} T^{20} - 29057164704 p^{5} T^{21} + 146376772 p^{6} T^{22} + 257608992 p^{7} T^{23} + 94591561 p^{8} T^{24} + 14111412 p^{9} T^{25} + 2390350 p^{10} T^{26} + 267528 p^{11} T^{27} + 33659 p^{12} T^{28} + 2712 p^{13} T^{29} + 274 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \)
61 \( 1 + 2 T - 284 T^{2} - 532 T^{3} + 39027 T^{4} + 1040 p T^{5} - 3708304 T^{6} - 4676926 T^{7} + 296804021 T^{8} + 193740812 T^{9} - 22216865364 T^{10} + 2263865252 T^{11} + 26083131090 p T^{12} - 777681597180 T^{13} - 107061062698164 T^{14} + 27225233412132 T^{15} + 6735249327156178 T^{16} + 27225233412132 p T^{17} - 107061062698164 p^{2} T^{18} - 777681597180 p^{3} T^{19} + 26083131090 p^{5} T^{20} + 2263865252 p^{5} T^{21} - 22216865364 p^{6} T^{22} + 193740812 p^{7} T^{23} + 296804021 p^{8} T^{24} - 4676926 p^{9} T^{25} - 3708304 p^{10} T^{26} + 1040 p^{12} T^{27} + 39027 p^{12} T^{28} - 532 p^{13} T^{29} - 284 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \)
67 \( 1 - 6 T + 286 T^{2} - 1644 T^{3} + 43631 T^{4} - 236568 T^{5} + 4737766 T^{6} - 22990638 T^{7} + 398969617 T^{8} - 1610111892 T^{9} + 27100830904 T^{10} - 72939944868 T^{11} + 1550073309166 T^{12} - 1152618024240 T^{13} + 82643397212316 T^{14} + 96791268580596 T^{15} + 4916320402820562 T^{16} + 96791268580596 p T^{17} + 82643397212316 p^{2} T^{18} - 1152618024240 p^{3} T^{19} + 1550073309166 p^{4} T^{20} - 72939944868 p^{5} T^{21} + 27100830904 p^{6} T^{22} - 1610111892 p^{7} T^{23} + 398969617 p^{8} T^{24} - 22990638 p^{9} T^{25} + 4737766 p^{10} T^{26} - 236568 p^{11} T^{27} + 43631 p^{12} T^{28} - 1644 p^{13} T^{29} + 286 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \)
71 \( 1 + 72 T + 2866 T^{2} + 81936 T^{3} + 1874819 T^{4} + 36376608 T^{5} + 619872622 T^{6} + 9493315848 T^{7} + 132781962265 T^{8} + 1716163196016 T^{9} + 20675777793940 T^{10} + 233712376216752 T^{11} + 2490665482595230 T^{12} + 25112799419069760 T^{13} + 240161804795734728 T^{14} + 2181997677737024976 T^{15} + 18851591748533517618 T^{16} + 2181997677737024976 p T^{17} + 240161804795734728 p^{2} T^{18} + 25112799419069760 p^{3} T^{19} + 2490665482595230 p^{4} T^{20} + 233712376216752 p^{5} T^{21} + 20675777793940 p^{6} T^{22} + 1716163196016 p^{7} T^{23} + 132781962265 p^{8} T^{24} + 9493315848 p^{9} T^{25} + 619872622 p^{10} T^{26} + 36376608 p^{11} T^{27} + 1874819 p^{12} T^{28} + 81936 p^{13} T^{29} + 2866 p^{14} T^{30} + 72 p^{15} T^{31} + p^{16} T^{32} \)
73 \( 1 - 624 T^{2} + 199300 T^{4} - 43237536 T^{6} + 7121591556 T^{8} - 942220990848 T^{10} + 103353082816700 T^{12} - 9571774645958256 T^{14} + 755662117040708342 T^{16} - 9571774645958256 p^{2} T^{18} + 103353082816700 p^{4} T^{20} - 942220990848 p^{6} T^{22} + 7121591556 p^{8} T^{24} - 43237536 p^{10} T^{26} + 199300 p^{12} T^{28} - 624 p^{14} T^{30} + p^{16} T^{32} \)
79 \( ( 1 + 18 T + 534 T^{2} + 6630 T^{3} + 118228 T^{4} + 1167414 T^{5} + 16071210 T^{6} + 133361058 T^{7} + 1512854550 T^{8} + 133361058 p T^{9} + 16071210 p^{2} T^{10} + 1167414 p^{3} T^{11} + 118228 p^{4} T^{12} + 6630 p^{5} T^{13} + 534 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
83 \( 1 - 1152 T^{2} + 633236 T^{4} - 220579696 T^{6} + 54522090212 T^{8} - 10143559934384 T^{10} + 1469728075460332 T^{12} - 169208570584982496 T^{14} + 15645267807767106550 T^{16} - 169208570584982496 p^{2} T^{18} + 1469728075460332 p^{4} T^{20} - 10143559934384 p^{6} T^{22} + 54522090212 p^{8} T^{24} - 220579696 p^{10} T^{26} + 633236 p^{12} T^{28} - 1152 p^{14} T^{30} + p^{16} T^{32} \)
89 \( 1 - 24 T + 496 T^{2} - 7296 T^{3} + 93914 T^{4} - 1030992 T^{5} + 9762672 T^{6} - 80684208 T^{7} + 554859669 T^{8} - 3056761272 T^{9} + 15037213224 T^{10} - 115686599640 T^{11} + 2115781897618 T^{12} - 39434796804024 T^{13} + 564212255027384 T^{14} - 6943105470611808 T^{15} + 68855149174833212 T^{16} - 6943105470611808 p T^{17} + 564212255027384 p^{2} T^{18} - 39434796804024 p^{3} T^{19} + 2115781897618 p^{4} T^{20} - 115686599640 p^{5} T^{21} + 15037213224 p^{6} T^{22} - 3056761272 p^{7} T^{23} + 554859669 p^{8} T^{24} - 80684208 p^{9} T^{25} + 9762672 p^{10} T^{26} - 1030992 p^{11} T^{27} + 93914 p^{12} T^{28} - 7296 p^{13} T^{29} + 496 p^{14} T^{30} - 24 p^{15} T^{31} + p^{16} T^{32} \)
97 \( 1 - 60 T + 2288 T^{2} - 65280 T^{3} + 1538491 T^{4} - 31004376 T^{5} + 549992224 T^{6} - 8713246332 T^{7} + 124868230421 T^{8} - 1630940606808 T^{9} + 19569122751704 T^{10} - 217157160901488 T^{11} + 2250938270934434 T^{12} - 22093904710368648 T^{13} + 209899289433638488 T^{14} - 1984773597050683584 T^{15} + 19210705001366999914 T^{16} - 1984773597050683584 p T^{17} + 209899289433638488 p^{2} T^{18} - 22093904710368648 p^{3} T^{19} + 2250938270934434 p^{4} T^{20} - 217157160901488 p^{5} T^{21} + 19569122751704 p^{6} T^{22} - 1630940606808 p^{7} T^{23} + 124868230421 p^{8} T^{24} - 8713246332 p^{9} T^{25} + 549992224 p^{10} T^{26} - 31004376 p^{11} T^{27} + 1538491 p^{12} T^{28} - 65280 p^{13} T^{29} + 2288 p^{14} T^{30} - 60 p^{15} T^{31} + p^{16} T^{32} \)
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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

−2.56592253154396949762769617525, −2.45455197369202445891134557804, −2.36201059887819938313093607028, −2.20988289246423149985550880488, −2.08232664344585830615159807868, −2.06886263005888142499309461867, −2.01995275234241434370369402747, −1.99650109803232058104758008020, −1.95457308790070411542983616614, −1.94902640714682182589886758903, −1.85311969931443936611856575249, −1.78935255490155753565446788957, −1.61765450099762429672020880744, −1.60546912566357794568833747138, −1.42426017487673592709458939895, −1.40193820636089949636065430394, −1.38705145979672824122522194713, −1.30377955131180550174603021177, −1.20808483565098098752016492110, −0.66089757776636581222029026397, −0.64241284029492849360168968923, −0.44304894941775728190220997553, −0.20682438651421476396378786095, −0.06730994294964929536256026661, −0.06494104212940235641583253764, 0.06494104212940235641583253764, 0.06730994294964929536256026661, 0.20682438651421476396378786095, 0.44304894941775728190220997553, 0.64241284029492849360168968923, 0.66089757776636581222029026397, 1.20808483565098098752016492110, 1.30377955131180550174603021177, 1.38705145979672824122522194713, 1.40193820636089949636065430394, 1.42426017487673592709458939895, 1.60546912566357794568833747138, 1.61765450099762429672020880744, 1.78935255490155753565446788957, 1.85311969931443936611856575249, 1.94902640714682182589886758903, 1.95457308790070411542983616614, 1.99650109803232058104758008020, 2.01995275234241434370369402747, 2.06886263005888142499309461867, 2.08232664344585830615159807868, 2.20988289246423149985550880488, 2.36201059887819938313093607028, 2.45455197369202445891134557804, 2.56592253154396949762769617525

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

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