Properties

Degree $8$
Conductor $2085136$
Sign $1$
Motivic weight $9$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 64·2-s + 84·3-s + 2.56e3·4-s − 1.39e3·5-s − 5.37e3·6-s + 1.23e4·7-s − 8.19e4·8-s − 2.75e4·9-s + 8.92e4·10-s − 1.04e5·11-s + 2.15e5·12-s + 1.20e5·13-s − 7.87e5·14-s − 1.17e5·15-s + 2.29e6·16-s − 4.12e5·17-s + 1.76e6·18-s + 5.21e5·19-s − 3.57e6·20-s + 1.03e6·21-s + 6.67e6·22-s + 3.01e6·23-s − 6.88e6·24-s + 1.94e6·25-s − 7.71e6·26-s + 1.06e6·27-s + 3.15e7·28-s + ⋯
L(s)  = 1  − 2.82·2-s + 0.598·3-s + 5·4-s − 0.998·5-s − 1.69·6-s + 1.93·7-s − 7.07·8-s − 1.40·9-s + 2.82·10-s − 2.14·11-s + 2.99·12-s + 1.17·13-s − 5.47·14-s − 0.597·15-s + 35/4·16-s − 1.19·17-s + 3.96·18-s + 0.917·19-s − 4.99·20-s + 1.15·21-s + 6.07·22-s + 2.24·23-s − 4.23·24-s + 0.996·25-s − 3.30·26-s + 0.385·27-s + 9.68·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2085136\)    =    \(2^{4} \cdot 19^{4}\)
Sign: $1$
Motivic weight: \(9\)
Character: induced by $\chi_{38} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2085136,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(1.15634\)
\(L(\frac12)\) \(\approx\) \(1.15634\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + p^{4} T )^{4} \)
19$C_1$ \( ( 1 - p^{4} T )^{4} \)
good3$C_2 \wr S_4$ \( 1 - 28 p T + 34625 T^{2} - 2096378 p T^{3} + 72317420 p^{2} T^{4} - 2096378 p^{10} T^{5} + 34625 p^{18} T^{6} - 28 p^{28} T^{7} + p^{36} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 279 p T - 206 p T^{2} + 65059389 p^{2} T^{3} + 62045170554 p^{3} T^{4} + 65059389 p^{11} T^{5} - 206 p^{19} T^{6} + 279 p^{28} T^{7} + p^{36} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 12307 T + 146953147 T^{2} - 177878821772 p T^{3} + 182915235883424 p^{2} T^{4} - 177878821772 p^{10} T^{5} + 146953147 p^{18} T^{6} - 12307 p^{27} T^{7} + p^{36} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 104249 T + 12830631428 T^{2} + 70344030474447 p T^{3} + 49503846191918961558 T^{4} + 70344030474447 p^{10} T^{5} + 12830631428 p^{18} T^{6} + 104249 p^{27} T^{7} + p^{36} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 120486 T + 11611428199 T^{2} - 1537985136135464 T^{3} + \)\(26\!\cdots\!96\)\( T^{4} - 1537985136135464 p^{9} T^{5} + 11611428199 p^{18} T^{6} - 120486 p^{27} T^{7} + p^{36} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 412139 T + 99978579665 T^{2} + 47122140457475886 T^{3} + \)\(22\!\cdots\!54\)\( T^{4} + 47122140457475886 p^{9} T^{5} + 99978579665 p^{18} T^{6} + 412139 p^{27} T^{7} + p^{36} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 3010300 T + 8067019202447 T^{2} - 26728667320812612 p^{2} T^{3} + \)\(22\!\cdots\!92\)\( T^{4} - 26728667320812612 p^{11} T^{5} + 8067019202447 p^{18} T^{6} - 3010300 p^{27} T^{7} + p^{36} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 6153240 T + 66652695791987 T^{2} - \)\(26\!\cdots\!04\)\( T^{3} + \)\(15\!\cdots\!08\)\( T^{4} - \)\(26\!\cdots\!04\)\( p^{9} T^{5} + 66652695791987 p^{18} T^{6} - 6153240 p^{27} T^{7} + p^{36} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 12774024 T + 157641860418028 T^{2} - \)\(10\!\cdots\!12\)\( T^{3} + \)\(69\!\cdots\!98\)\( T^{4} - \)\(10\!\cdots\!12\)\( p^{9} T^{5} + 157641860418028 p^{18} T^{6} - 12774024 p^{27} T^{7} + p^{36} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 20506048 T + 451640199111340 T^{2} - \)\(57\!\cdots\!32\)\( T^{3} + \)\(77\!\cdots\!30\)\( T^{4} - \)\(57\!\cdots\!32\)\( p^{9} T^{5} + 451640199111340 p^{18} T^{6} - 20506048 p^{27} T^{7} + p^{36} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 11620300 T + 777621524436344 T^{2} - \)\(46\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!26\)\( T^{4} - \)\(46\!\cdots\!00\)\( p^{9} T^{5} + 777621524436344 p^{18} T^{6} - 11620300 p^{27} T^{7} + p^{36} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 7698327 T + 1116374999600644 T^{2} - \)\(20\!\cdots\!79\)\( T^{3} + \)\(59\!\cdots\!74\)\( T^{4} - \)\(20\!\cdots\!79\)\( p^{9} T^{5} + 1116374999600644 p^{18} T^{6} - 7698327 p^{27} T^{7} + p^{36} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 31581083 T + 59729788569388 p T^{2} + \)\(86\!\cdots\!23\)\( T^{3} + \)\(43\!\cdots\!46\)\( T^{4} + \)\(86\!\cdots\!23\)\( p^{9} T^{5} + 59729788569388 p^{19} T^{6} + 31581083 p^{27} T^{7} + p^{36} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 72549422 T + 9181124351782847 T^{2} - \)\(40\!\cdots\!96\)\( T^{3} + \)\(36\!\cdots\!60\)\( T^{4} - \)\(40\!\cdots\!96\)\( p^{9} T^{5} + 9181124351782847 p^{18} T^{6} - 72549422 p^{27} T^{7} + p^{36} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 149234120 T + 38463872258171009 T^{2} + \)\(37\!\cdots\!70\)\( T^{3} + \)\(51\!\cdots\!56\)\( T^{4} + \)\(37\!\cdots\!70\)\( p^{9} T^{5} + 38463872258171009 p^{18} T^{6} + 149234120 p^{27} T^{7} + p^{36} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 129004373 T + 36134043355765018 T^{2} - \)\(36\!\cdots\!19\)\( T^{3} + \)\(61\!\cdots\!14\)\( T^{4} - \)\(36\!\cdots\!19\)\( p^{9} T^{5} + 36134043355765018 p^{18} T^{6} - 129004373 p^{27} T^{7} + p^{36} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 132595266 T + 50476271812135045 T^{2} - \)\(24\!\cdots\!06\)\( T^{3} + \)\(10\!\cdots\!12\)\( T^{4} - \)\(24\!\cdots\!06\)\( p^{9} T^{5} + 50476271812135045 p^{18} T^{6} - 132595266 p^{27} T^{7} + p^{36} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 47138482 T + 109274701906688168 T^{2} - \)\(43\!\cdots\!18\)\( T^{3} + \)\(54\!\cdots\!62\)\( T^{4} - \)\(43\!\cdots\!18\)\( p^{9} T^{5} + 109274701906688168 p^{18} T^{6} + 47138482 p^{27} T^{7} + p^{36} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 39332795 T + 147831632963514661 T^{2} - \)\(30\!\cdots\!82\)\( T^{3} + \)\(10\!\cdots\!82\)\( T^{4} - \)\(30\!\cdots\!82\)\( p^{9} T^{5} + 147831632963514661 p^{18} T^{6} + 39332795 p^{27} T^{7} + p^{36} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 307010840 T + 250097675403470728 T^{2} + \)\(12\!\cdots\!20\)\( T^{3} + \)\(34\!\cdots\!42\)\( T^{4} + \)\(12\!\cdots\!20\)\( p^{9} T^{5} + 250097675403470728 p^{18} T^{6} + 307010840 p^{27} T^{7} + p^{36} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 746568232 T + 900563380389680840 T^{2} + \)\(42\!\cdots\!92\)\( T^{3} + \)\(26\!\cdots\!54\)\( T^{4} + \)\(42\!\cdots\!92\)\( p^{9} T^{5} + 900563380389680840 p^{18} T^{6} + 746568232 p^{27} T^{7} + p^{36} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 286943482 T + 949493363525023412 T^{2} - \)\(20\!\cdots\!34\)\( T^{3} + \)\(42\!\cdots\!38\)\( T^{4} - \)\(20\!\cdots\!34\)\( p^{9} T^{5} + 949493363525023412 p^{18} T^{6} - 286943482 p^{27} T^{7} + p^{36} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 793519958 T - 696016064440921328 T^{2} - \)\(23\!\cdots\!78\)\( T^{3} + \)\(15\!\cdots\!70\)\( T^{4} - \)\(23\!\cdots\!78\)\( p^{9} T^{5} - 696016064440921328 p^{18} T^{6} - 793519958 p^{27} T^{7} + p^{36} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.34570591117792494774103085855, −9.687027440152370595237789606900, −9.129828211633625094292699040503, −8.997917585349156301939243383511, −8.643518667869504223382819281776, −8.356835922583964724022482897467, −8.191904380635449026958655625272, −7.893294801999888533030083100752, −7.86442293455574043817575950135, −7.39046153947799109496659904811, −6.80372721054149361834684572273, −6.42444062838758251134464802650, −6.27182929119480261575562545553, −5.33803547761509390244254760403, −5.09102612393007104888397543886, −4.77690315618524440666991753821, −4.14989855545342682112772739389, −3.00418316438181926951109022527, −2.95073188165454631225153947198, −2.63360750445587434446796692078, −2.40668740999092649516449211382, −1.30234077860475523544203894450, −1.21813914751753757329603681342, −0.71196895381803246962145644293, −0.38491129285373576536360182428, 0.38491129285373576536360182428, 0.71196895381803246962145644293, 1.21813914751753757329603681342, 1.30234077860475523544203894450, 2.40668740999092649516449211382, 2.63360750445587434446796692078, 2.95073188165454631225153947198, 3.00418316438181926951109022527, 4.14989855545342682112772739389, 4.77690315618524440666991753821, 5.09102612393007104888397543886, 5.33803547761509390244254760403, 6.27182929119480261575562545553, 6.42444062838758251134464802650, 6.80372721054149361834684572273, 7.39046153947799109496659904811, 7.86442293455574043817575950135, 7.893294801999888533030083100752, 8.191904380635449026958655625272, 8.356835922583964724022482897467, 8.643518667869504223382819281776, 8.997917585349156301939243383511, 9.129828211633625094292699040503, 9.687027440152370595237789606900, 10.34570591117792494774103085855

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