Properties

Label 8-74e4-1.1-c3e4-0-0
Degree $8$
Conductor $29986576$
Sign $1$
Analytic cond. $363.405$
Root an. cond. $2.08953$
Motivic weight $3$
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  + 8·2-s + 4·3-s + 40·4-s + 21·5-s + 32·6-s + 23·7-s + 160·8-s − 2·9-s + 168·10-s − 66·11-s + 160·12-s + 53·13-s + 184·14-s + 84·15-s + 560·16-s + 12·17-s − 16·18-s − 34·19-s + 840·20-s + 92·21-s − 528·22-s + 45·23-s + 640·24-s − 110·25-s + 424·26-s − 231·27-s + 920·28-s + ⋯
L(s)  = 1  + 2.82·2-s + 0.769·3-s + 5·4-s + 1.87·5-s + 2.17·6-s + 1.24·7-s + 7.07·8-s − 0.0740·9-s + 5.31·10-s − 1.80·11-s + 3.84·12-s + 1.13·13-s + 3.51·14-s + 1.44·15-s + 35/4·16-s + 0.171·17-s − 0.209·18-s − 0.410·19-s + 9.39·20-s + 0.956·21-s − 5.11·22-s + 0.407·23-s + 5.44·24-s − 0.879·25-s + 3.19·26-s − 1.64·27-s + 6.20·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29986576 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29986576 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(29986576\)    =    \(2^{4} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(363.405\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 29986576,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(35.07222658\)
\(L(\frac12)\) \(\approx\) \(35.07222658\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{4} \)
37$C_1$ \( ( 1 - p T )^{4} \)
good3$C_2 \wr S_4$ \( 1 - 4 T + 2 p^{2} T^{2} + 151 T^{3} - 538 T^{4} + 151 p^{3} T^{5} + 2 p^{8} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 21 T + 551 T^{2} - 6629 T^{3} + 3996 p^{2} T^{4} - 6629 p^{3} T^{5} + 551 p^{6} T^{6} - 21 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 23 T + 334 T^{2} + 3321 T^{3} - 82974 T^{4} + 3321 p^{3} T^{5} + 334 p^{6} T^{6} - 23 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 6 p T + 3050 T^{2} + 10103 p T^{3} + 4343022 T^{4} + 10103 p^{4} T^{5} + 3050 p^{6} T^{6} + 6 p^{10} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 53 T + 7135 T^{2} - 287013 T^{3} + 21453756 T^{4} - 287013 p^{3} T^{5} + 7135 p^{6} T^{6} - 53 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 12 T + 8564 T^{2} - 71988 T^{3} + 2243190 p T^{4} - 71988 p^{3} T^{5} + 8564 p^{6} T^{6} - 12 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 34 T + 12376 T^{2} + 159322 T^{3} + 86762206 T^{4} + 159322 p^{3} T^{5} + 12376 p^{6} T^{6} + 34 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 45 T + 16763 T^{2} + 730449 T^{3} + 68493696 T^{4} + 730449 p^{3} T^{5} + 16763 p^{6} T^{6} - 45 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 21 T + 60737 T^{2} + 4521531 T^{3} + 1713089724 T^{4} + 4521531 p^{3} T^{5} + 60737 p^{6} T^{6} + 21 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 17 T + 83059 T^{2} + 723299 T^{3} + 3138849512 T^{4} + 723299 p^{3} T^{5} + 83059 p^{6} T^{6} - 17 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 174 T + 176018 T^{2} + 45396789 T^{3} + 14556018876 T^{4} + 45396789 p^{3} T^{5} + 176018 p^{6} T^{6} + 174 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 514 T + 272248 T^{2} + 78378674 T^{3} + 26122615550 T^{4} + 78378674 p^{3} T^{5} + 272248 p^{6} T^{6} + 514 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 93 T + 243062 T^{2} + 12602917 T^{3} + 32415329394 T^{4} + 12602917 p^{3} T^{5} + 243062 p^{6} T^{6} + 93 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 3 T + 250394 T^{2} + 104921751 T^{3} + 21283475370 T^{4} + 104921751 p^{3} T^{5} + 250394 p^{6} T^{6} - 3 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 6 p T + 864224 T^{2} - 220122962 T^{3} + 270657625134 T^{4} - 220122962 p^{3} T^{5} + 864224 p^{6} T^{6} - 6 p^{10} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 1139 T + 1052815 T^{2} - 701819743 T^{3} + 361620950012 T^{4} - 701819743 p^{3} T^{5} + 1052815 p^{6} T^{6} - 1139 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 965 T + 1405201 T^{2} - 857178861 T^{3} + 659618123892 T^{4} - 857178861 p^{3} T^{5} + 1405201 p^{6} T^{6} - 965 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 27 T + 1359572 T^{2} - 25971027 T^{3} + 717123322230 T^{4} - 25971027 p^{3} T^{5} + 1359572 p^{6} T^{6} - 27 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 2168 T + 3199948 T^{2} - 3058097161 T^{3} + 2248284604112 T^{4} - 3058097161 p^{3} T^{5} + 3199948 p^{6} T^{6} - 2168 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 449 T + 979099 T^{2} - 819844635 T^{3} + 489483284352 T^{4} - 819844635 p^{3} T^{5} + 979099 p^{6} T^{6} - 449 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 1293 T + 2707820 T^{2} + 2247238189 T^{3} + 2441373917718 T^{4} + 2247238189 p^{3} T^{5} + 2707820 p^{6} T^{6} + 1293 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 264 T + 1806680 T^{2} + 419192104 T^{3} + 1814445938382 T^{4} + 419192104 p^{3} T^{5} + 1806680 p^{6} T^{6} + 264 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 842 T + 2807848 T^{2} - 2129523030 T^{3} + 3464925879534 T^{4} - 2129523030 p^{3} T^{5} + 2807848 p^{6} T^{6} - 842 p^{9} T^{7} + p^{12} T^{8} \)
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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

−10.29405099157376687273412147423, −10.24778440034677909657849119491, −9.708020540113319345878004714835, −9.592356125945876713254049438119, −9.333898564470110507953576634426, −8.350406552697053301663673365327, −8.297439019430225493316775507062, −8.088484147158937699201083407132, −7.936441830182450231785779155032, −7.29901705363922440616977942070, −6.94597415030121571555139414599, −6.45588116433919486410873472989, −6.37176639925715880742506295631, −5.70214596633901270947942535338, −5.50343274454870716552099385767, −5.41351776109249466082860247921, −5.15128604280149089752487640529, −4.65862504702484858646454520761, −3.92588239213037693408896524305, −3.83942942557648481900165550905, −3.35327857471839286844546667230, −2.59790012798435350382181632576, −2.26998011600194616729009763662, −2.02630962582907765948964943083, −1.49522337671883843159128814768, 1.49522337671883843159128814768, 2.02630962582907765948964943083, 2.26998011600194616729009763662, 2.59790012798435350382181632576, 3.35327857471839286844546667230, 3.83942942557648481900165550905, 3.92588239213037693408896524305, 4.65862504702484858646454520761, 5.15128604280149089752487640529, 5.41351776109249466082860247921, 5.50343274454870716552099385767, 5.70214596633901270947942535338, 6.37176639925715880742506295631, 6.45588116433919486410873472989, 6.94597415030121571555139414599, 7.29901705363922440616977942070, 7.936441830182450231785779155032, 8.088484147158937699201083407132, 8.297439019430225493316775507062, 8.350406552697053301663673365327, 9.333898564470110507953576634426, 9.592356125945876713254049438119, 9.708020540113319345878004714835, 10.24778440034677909657849119491, 10.29405099157376687273412147423

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