Properties

Degree $8$
Conductor $4.741\times 10^{13}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·5-s + 2·9-s − 4·11-s + 8·15-s − 4·17-s − 6·19-s − 12·23-s + 4·25-s + 4·27-s + 4·29-s − 8·31-s + 8·33-s − 16·37-s − 4·41-s − 4·43-s − 8·45-s − 6·47-s − 6·49-s + 8·51-s + 16·53-s + 16·55-s + 12·57-s − 12·59-s − 24·61-s − 28·67-s + 24·69-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s + 2/3·9-s − 1.20·11-s + 2.06·15-s − 0.970·17-s − 1.37·19-s − 2.50·23-s + 4/5·25-s + 0.769·27-s + 0.742·29-s − 1.43·31-s + 1.39·33-s − 2.63·37-s − 0.624·41-s − 0.609·43-s − 1.19·45-s − 0.875·47-s − 6/7·49-s + 1.12·51-s + 2.19·53-s + 2.15·55-s + 1.58·57-s − 1.56·59-s − 3.07·61-s − 3.42·67-s + 2.88·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 41^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{2624} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{24} \cdot 41^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
41$C_1$ \( ( 1 + T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + 2 T + 2 T^{2} - 4 T^{3} - 8 T^{4} - 4 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 4 T + 12 T^{2} + 16 T^{3} + 34 T^{4} + 16 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 6 T^{2} + 26 T^{3} + 24 T^{4} + 26 p T^{5} + 6 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 4 T + 26 T^{2} + 114 T^{3} + 384 T^{4} + 114 p T^{5} + 26 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 12 T^{2} + 48 T^{3} + 118 T^{4} + 48 p T^{5} + 12 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 4 T + 20 T^{2} + 124 T^{3} + 534 T^{4} + 124 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 6 T + 62 T^{2} + 208 T^{3} + 1448 T^{4} + 208 p T^{5} + 62 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 12 T + 108 T^{2} + 700 T^{3} + 3718 T^{4} + 700 p T^{5} + 108 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 4 T + 76 T^{2} - 12 p T^{3} + 2870 T^{4} - 12 p^{2} T^{5} + 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 8 T + 92 T^{2} + 712 T^{3} + 3846 T^{4} + 712 p T^{5} + 92 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 16 T + 212 T^{2} + 1740 T^{3} + 12626 T^{4} + 1740 p T^{5} + 212 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 4 T + 124 T^{2} + 244 T^{3} + 6678 T^{4} + 244 p T^{5} + 124 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 6 T + 126 T^{2} + 640 T^{3} + 8608 T^{4} + 640 p T^{5} + 126 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 16 T + 4 p T^{2} - 1824 T^{3} + 15558 T^{4} - 1824 p T^{5} + 4 p^{3} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 12 T + 252 T^{2} + 1996 T^{3} + 22582 T^{4} + 1996 p T^{5} + 252 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 24 T + 420 T^{2} + 4824 T^{3} + 44086 T^{4} + 4824 p T^{5} + 420 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 28 T + 538 T^{2} + 6638 T^{3} + 64208 T^{4} + 6638 p T^{5} + 538 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 2 T + 98 T^{2} - 268 T^{3} + 48 p T^{4} - 268 p T^{5} + 98 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 8 T + 212 T^{2} - 1060 T^{3} + 19890 T^{4} - 1060 p T^{5} + 212 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 18 T + 366 T^{2} + 4224 T^{3} + 45328 T^{4} + 4224 p T^{5} + 366 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 12 T + 252 T^{2} - 1644 T^{3} + 24598 T^{4} - 1644 p T^{5} + 252 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 4 T + 228 T^{2} - 796 T^{3} + 326 p T^{4} - 796 p T^{5} + 228 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 16 T + 268 T^{2} - 3376 T^{3} + 38118 T^{4} - 3376 p T^{5} + 268 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.58736806433065963291962686812, −6.57875269723102083793665804859, −6.22657938405028622370468691207, −6.10929513368922646515655666244, −6.00880883878907367762272700938, −5.65223751121941573318095230086, −5.38969419707564642242373279431, −5.30502774516938323048043522920, −4.92524220134133991286090223038, −4.89373688391566700654953551599, −4.62959974407541175858918861065, −4.42732382373613428576717511126, −4.21398649051909834752450468125, −3.97700276125369228605302336365, −3.86231700376536190279606304038, −3.64246933599559843307527313298, −3.30770383722639828433509057974, −3.00370275110694731496600103693, −2.87842671206415468637803350058, −2.64521256393266791491561857519, −2.07959999227845622141877470536, −1.95366203435193979692186451947, −1.63788858870495298883117733410, −1.55545745047857189007662002067, −0.919157258962756312710999535339, 0, 0, 0, 0, 0.919157258962756312710999535339, 1.55545745047857189007662002067, 1.63788858870495298883117733410, 1.95366203435193979692186451947, 2.07959999227845622141877470536, 2.64521256393266791491561857519, 2.87842671206415468637803350058, 3.00370275110694731496600103693, 3.30770383722639828433509057974, 3.64246933599559843307527313298, 3.86231700376536190279606304038, 3.97700276125369228605302336365, 4.21398649051909834752450468125, 4.42732382373613428576717511126, 4.62959974407541175858918861065, 4.89373688391566700654953551599, 4.92524220134133991286090223038, 5.30502774516938323048043522920, 5.38969419707564642242373279431, 5.65223751121941573318095230086, 6.00880883878907367762272700938, 6.10929513368922646515655666244, 6.22657938405028622370468691207, 6.57875269723102083793665804859, 6.58736806433065963291962686812

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