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Properties

Label 8-1840e4-1.1-c3e4-0-2
Degree $8$
Conductor $1.146\times 10^{13}$
Sign $1$
Analytic cond. $1.38910\times 10^{8}$
Root an. cond. $10.4193$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 14·3-s + 20·5-s − 8·7-s + 76·9-s − 21·11-s + 70·13-s − 280·15-s + 56·17-s − 173·19-s + 112·21-s + 92·23-s + 250·25-s − 153·27-s − 118·29-s − 17·31-s + 294·33-s − 160·35-s − 343·37-s − 980·39-s + 139·41-s + 50·43-s + 1.52e3·45-s − 367·47-s − 716·49-s − 784·51-s − 353·53-s − 420·55-s + ⋯
L(s)  = 1  − 2.69·3-s + 1.78·5-s − 0.431·7-s + 2.81·9-s − 0.575·11-s + 1.49·13-s − 4.81·15-s + 0.798·17-s − 2.08·19-s + 1.16·21-s + 0.834·23-s + 2·25-s − 1.09·27-s − 0.755·29-s − 0.0984·31-s + 1.55·33-s − 0.772·35-s − 1.52·37-s − 4.02·39-s + 0.529·41-s + 0.177·43-s + 5.03·45-s − 1.13·47-s − 2.08·49-s − 2.15·51-s − 0.914·53-s − 1.02·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(1.38910\times 10^{8}\)
Root analytic conductor: \(10.4193\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 23^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - p T )^{4} \)
23$C_1$ \( ( 1 - p T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + 14 T + 40 p T^{2} + 769 T^{3} + 4118 T^{4} + 769 p^{3} T^{5} + 40 p^{7} T^{6} + 14 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 8 T + 780 T^{2} + 481 p T^{3} + 5932 p^{2} T^{4} + 481 p^{4} T^{5} + 780 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 21 T + 2719 T^{2} + 126987 T^{3} + 3530624 T^{4} + 126987 p^{3} T^{5} + 2719 p^{6} T^{6} + 21 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 70 T + 698 p T^{2} - 449961 T^{3} + 30171476 T^{4} - 449961 p^{3} T^{5} + 698 p^{7} T^{6} - 70 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 56 T + 11120 T^{2} - 560613 T^{3} + 68960382 T^{4} - 560613 p^{3} T^{5} + 11120 p^{6} T^{6} - 56 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 173 T + 25377 T^{2} + 2932999 T^{3} + 264013708 T^{4} + 2932999 p^{3} T^{5} + 25377 p^{6} T^{6} + 173 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 118 T + 61751 T^{2} + 8515068 T^{3} + 1867055088 T^{4} + 8515068 p^{3} T^{5} + 61751 p^{6} T^{6} + 118 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 17 T + 11172 T^{2} + 1640140 T^{3} + 1568834653 T^{4} + 1640140 p^{3} T^{5} + 11172 p^{6} T^{6} + 17 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 343 T + 102648 T^{2} + 19183121 T^{3} + 3172218526 T^{4} + 19183121 p^{3} T^{5} + 102648 p^{6} T^{6} + 343 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 139 T + 118058 T^{2} - 18482874 T^{3} + 11562193671 T^{4} - 18482874 p^{3} T^{5} + 118058 p^{6} T^{6} - 139 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 50 T + 110692 T^{2} + 2729918 T^{3} + 10322725382 T^{4} + 2729918 p^{3} T^{5} + 110692 p^{6} T^{6} - 50 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 367 T + 277058 T^{2} + 106977327 T^{3} + 36639696810 T^{4} + 106977327 p^{3} T^{5} + 277058 p^{6} T^{6} + 367 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 353 T + 483386 T^{2} + 121077363 T^{3} + 99311869194 T^{4} + 121077363 p^{3} T^{5} + 483386 p^{6} T^{6} + 353 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 453 T + 512870 T^{2} - 262193049 T^{3} + 135268242162 T^{4} - 262193049 p^{3} T^{5} + 512870 p^{6} T^{6} - 453 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 327 T + 703169 T^{2} + 181008321 T^{3} + 214272589464 T^{4} + 181008321 p^{3} T^{5} + 703169 p^{6} T^{6} + 327 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 455 T + 947140 T^{2} - 444464551 T^{3} + 387446061734 T^{4} - 444464551 p^{3} T^{5} + 947140 p^{6} T^{6} - 455 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 195 T + 725314 T^{2} + 106317378 T^{3} + 5198032027 p T^{4} + 106317378 p^{3} T^{5} + 725314 p^{6} T^{6} + 195 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 633 T + 18208 p T^{2} + 586224547 T^{3} + 715636791894 T^{4} + 586224547 p^{3} T^{5} + 18208 p^{7} T^{6} + 633 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 1140 T + 1445260 T^{2} - 1359377572 T^{3} + 999599980326 T^{4} - 1359377572 p^{3} T^{5} + 1445260 p^{6} T^{6} - 1140 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 1199 T + 2505156 T^{2} - 2065757331 T^{3} + 2211053981782 T^{4} - 2065757331 p^{3} T^{5} + 2505156 p^{6} T^{6} - 1199 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 2170 T + 3806124 T^{2} + 4376259102 T^{3} + 4366516459126 T^{4} + 4376259102 p^{3} T^{5} + 3806124 p^{6} T^{6} + 2170 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 703 T + 2942565 T^{2} + 1628352317 T^{3} + 3682446979948 T^{4} + 1628352317 p^{3} T^{5} + 2942565 p^{6} T^{6} + 703 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.52197094940141402904200653456, −6.25088516582552885632198862171, −6.21170109989493008445531989644, −6.03208341503351481574071643161, −5.85326096048897340685734598030, −5.46672274368700879520266645891, −5.38469431991405742697011336776, −5.32674599544431729628031137942, −5.19621120528347746749164415684, −4.77761914105980660528280090844, −4.69709942951543193643766580049, −4.46156721861264489297517485710, −4.04355783712721860671411314220, −3.79606800842726697104163130997, −3.48575860692944617139826485400, −3.45395955870093679787812028194, −3.09267279516497051140835331907, −2.64812085904445472517704256195, −2.39412411469810094416992007657, −2.37205323841963248678629554614, −1.83422905474787813760108246681, −1.59742102849379825935529347848, −1.26260744156583385940885165873, −1.09626332797714179193959630974, −0.989893487603366295089609356426, 0, 0, 0, 0, 0.989893487603366295089609356426, 1.09626332797714179193959630974, 1.26260744156583385940885165873, 1.59742102849379825935529347848, 1.83422905474787813760108246681, 2.37205323841963248678629554614, 2.39412411469810094416992007657, 2.64812085904445472517704256195, 3.09267279516497051140835331907, 3.45395955870093679787812028194, 3.48575860692944617139826485400, 3.79606800842726697104163130997, 4.04355783712721860671411314220, 4.46156721861264489297517485710, 4.69709942951543193643766580049, 4.77761914105980660528280090844, 5.19621120528347746749164415684, 5.32674599544431729628031137942, 5.38469431991405742697011336776, 5.46672274368700879520266645891, 5.85326096048897340685734598030, 6.03208341503351481574071643161, 6.21170109989493008445531989644, 6.25088516582552885632198862171, 6.52197094940141402904200653456

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