Properties

Label 8-38e4-1.1-c7e4-0-0
Degree $8$
Conductor $2085136$
Sign $1$
Analytic cond. $19856.1$
Root an. cond. $3.44537$
Motivic weight $7$
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  + 32·2-s + 12·3-s + 640·4-s − 279·5-s + 384·6-s + 2.48e3·7-s + 1.02e4·8-s + 439·9-s − 8.92e3·10-s + 5.26e3·11-s + 7.68e3·12-s + 5.40e3·13-s + 7.95e4·14-s − 3.34e3·15-s + 1.43e5·16-s + 2.28e4·17-s + 1.40e4·18-s + 2.74e4·19-s − 1.78e5·20-s + 2.98e4·21-s + 1.68e5·22-s + 3.36e3·23-s + 1.22e5·24-s − 6.10e4·25-s + 1.72e5·26-s − 7.74e4·27-s + 1.59e6·28-s + ⋯
L(s)  = 1  + 2.82·2-s + 0.256·3-s + 5·4-s − 0.998·5-s + 0.725·6-s + 2.73·7-s + 7.07·8-s + 0.200·9-s − 2.82·10-s + 1.19·11-s + 1.28·12-s + 0.682·13-s + 7.74·14-s − 0.256·15-s + 35/4·16-s + 1.12·17-s + 0.567·18-s + 0.917·19-s − 4.99·20-s + 0.702·21-s + 3.37·22-s + 0.0576·23-s + 1.81·24-s − 0.781·25-s + 1.93·26-s − 0.757·27-s + 13.6·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2085136\)    =    \(2^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(19856.1\)
Root analytic conductor: \(3.44537\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2085136,\ (\ :7/2, 7/2, 7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(55.89608886\)
\(L(\frac12)\) \(\approx\) \(55.89608886\)
\(L(\frac{9}{2})\) 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$C_1$ \( ( 1 - p^{3} T )^{4} \)
19$C_1$ \( ( 1 - p^{3} T )^{4} \)
good3$C_2 \wr S_4$ \( 1 - 4 p T - 295 T^{2} + 28754 p T^{3} - 100180 p^{2} T^{4} + 28754 p^{8} T^{5} - 295 p^{14} T^{6} - 4 p^{22} T^{7} + p^{28} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 279 T + 27778 p T^{2} + 1350081 p^{2} T^{3} + 25828818 p^{4} T^{4} + 1350081 p^{9} T^{5} + 27778 p^{15} T^{6} + 279 p^{21} T^{7} + p^{28} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 355 p T + 5045419 T^{2} - 6353668364 T^{3} + 6879076274144 T^{4} - 6353668364 p^{7} T^{5} + 5045419 p^{14} T^{6} - 355 p^{22} T^{7} + p^{28} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 479 p T + 63505124 T^{2} - 263253354585 T^{3} + 1803977512557558 T^{4} - 263253354585 p^{7} T^{5} + 63505124 p^{14} T^{6} - 479 p^{22} T^{7} + p^{28} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 5406 T + 7578475 p T^{2} + 402392756440 T^{3} + 1378699571519292 T^{4} + 402392756440 p^{7} T^{5} + 7578475 p^{15} T^{6} - 5406 p^{21} T^{7} + p^{28} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 22885 T + 739754033 T^{2} - 19792467663090 T^{3} + 456808640899749018 T^{4} - 19792467663090 p^{7} T^{5} + 739754033 p^{14} T^{6} - 22885 p^{21} T^{7} + p^{28} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 3364 T + 1546353407 T^{2} - 48695367446844 T^{3} + 16006630067785073616 T^{4} - 48695367446844 p^{7} T^{5} + 1546353407 p^{14} T^{6} - 3364 p^{21} T^{7} + p^{28} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 122136 T + 55100568923 T^{2} + 6538523897630628 T^{3} + \)\(13\!\cdots\!72\)\( T^{4} + 6538523897630628 p^{7} T^{5} + 55100568923 p^{14} T^{6} + 122136 p^{21} T^{7} + p^{28} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 225480 T + 69094969516 T^{2} - 15136676725778216 T^{3} + \)\(26\!\cdots\!38\)\( T^{4} - 15136676725778216 p^{7} T^{5} + 69094969516 p^{14} T^{6} - 225480 p^{21} T^{7} + p^{28} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 154096 T + 43964761900 T^{2} + 22759340129468464 T^{3} + \)\(17\!\cdots\!50\)\( T^{4} + 22759340129468464 p^{7} T^{5} + 43964761900 p^{14} T^{6} - 154096 p^{21} T^{7} + p^{28} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 1054628 T + 1094991857624 T^{2} + 628358421549336204 T^{3} + \)\(34\!\cdots\!66\)\( T^{4} + 628358421549336204 p^{7} T^{5} + 1094991857624 p^{14} T^{6} + 1054628 p^{21} T^{7} + p^{28} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 840795 T + 879527106724 T^{2} + 530888279209871203 T^{3} + \)\(36\!\cdots\!94\)\( T^{4} + 530888279209871203 p^{7} T^{5} + 879527106724 p^{14} T^{6} + 840795 p^{21} T^{7} + p^{28} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 1021877 T + 1972473290900 T^{2} + 1286140928961756945 T^{3} + \)\(14\!\cdots\!82\)\( T^{4} + 1286140928961756945 p^{7} T^{5} + 1972473290900 p^{14} T^{6} + 1021877 p^{21} T^{7} + p^{28} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 326842 T + 4350954270263 T^{2} + 1116130028742868416 T^{3} + \)\(74\!\cdots\!72\)\( T^{4} + 1116130028742868416 p^{7} T^{5} + 4350954270263 p^{14} T^{6} + 326842 p^{21} T^{7} + p^{28} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 421384 T + 4416605620265 T^{2} + 5125206026271641286 T^{3} + \)\(62\!\cdots\!48\)\( T^{4} + 5125206026271641286 p^{7} T^{5} + 4416605620265 p^{14} T^{6} - 421384 p^{21} T^{7} + p^{28} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 116825 T + 9319855015114 T^{2} - 1161006564160465283 T^{3} + \)\(41\!\cdots\!66\)\( T^{4} - 1161006564160465283 p^{7} T^{5} + 9319855015114 p^{14} T^{6} - 116825 p^{21} T^{7} + p^{28} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 5794566 T + 35662430474173 T^{2} - \)\(11\!\cdots\!86\)\( T^{3} + \)\(36\!\cdots\!80\)\( T^{4} - \)\(11\!\cdots\!86\)\( p^{7} T^{5} + 35662430474173 p^{14} T^{6} - 5794566 p^{21} T^{7} + p^{28} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 10590626 T + 72774649146920 T^{2} - \)\(33\!\cdots\!66\)\( T^{3} + \)\(11\!\cdots\!82\)\( T^{4} - \)\(33\!\cdots\!66\)\( p^{7} T^{5} + 72774649146920 p^{14} T^{6} - 10590626 p^{21} T^{7} + p^{28} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 3971389 T + 11335631623573 T^{2} - 21295071158761180526 T^{3} + \)\(19\!\cdots\!02\)\( T^{4} - 21295071158761180526 p^{7} T^{5} + 11335631623573 p^{14} T^{6} - 3971389 p^{21} T^{7} + p^{28} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 5597800 T + 47755668016648 T^{2} - \)\(29\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!02\)\( T^{4} - \)\(29\!\cdots\!80\)\( p^{7} T^{5} + 47755668016648 p^{14} T^{6} - 5597800 p^{21} T^{7} + p^{28} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 4665800 T + 90191405937032 T^{2} - \)\(34\!\cdots\!20\)\( T^{3} + \)\(35\!\cdots\!58\)\( T^{4} - \)\(34\!\cdots\!20\)\( p^{7} T^{5} + 90191405937032 p^{14} T^{6} - 4665800 p^{21} T^{7} + p^{28} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 2794214 T + 98763681921620 T^{2} + \)\(34\!\cdots\!66\)\( T^{3} + \)\(50\!\cdots\!58\)\( T^{4} + \)\(34\!\cdots\!66\)\( p^{7} T^{5} + 98763681921620 p^{14} T^{6} + 2794214 p^{21} T^{7} + p^{28} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 14445130 T + 287388834797392 T^{2} + \)\(30\!\cdots\!06\)\( T^{3} + \)\(33\!\cdots\!74\)\( T^{4} + \)\(30\!\cdots\!06\)\( p^{7} T^{5} + 287388834797392 p^{14} T^{6} + 14445130 p^{21} T^{7} + p^{28} T^{8} \)
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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

−11.10712776078351919528325205285, −10.69259498468142845607361922200, −10.05032404959926932094125265068, −9.676521142858511169076551990972, −9.575313490721812167918691502299, −8.504177751150700582470175843934, −8.295448014419135264751998609793, −7.918909106248510156017119889199, −7.86795451400915874137527167136, −7.58415274276687506447620182506, −6.73800115580524426060838032892, −6.60870133113669648643913424612, −6.37350989514509866107127305020, −5.46608722085194530449009074179, −5.23904749971792351459217545098, −5.08163851258001102960551192924, −4.66574625484201104346814702666, −4.19889929609670336305552458801, −3.65047278078697587855394361871, −3.55231470337234971894709657575, −3.22922226701356345782655227584, −1.98206711207948506378978435290, −1.93786134775705376739493591476, −1.41991074334186519796930473037, −0.857478437018250913227287613304, 0.857478437018250913227287613304, 1.41991074334186519796930473037, 1.93786134775705376739493591476, 1.98206711207948506378978435290, 3.22922226701356345782655227584, 3.55231470337234971894709657575, 3.65047278078697587855394361871, 4.19889929609670336305552458801, 4.66574625484201104346814702666, 5.08163851258001102960551192924, 5.23904749971792351459217545098, 5.46608722085194530449009074179, 6.37350989514509866107127305020, 6.60870133113669648643913424612, 6.73800115580524426060838032892, 7.58415274276687506447620182506, 7.86795451400915874137527167136, 7.918909106248510156017119889199, 8.295448014419135264751998609793, 8.504177751150700582470175843934, 9.575313490721812167918691502299, 9.676521142858511169076551990972, 10.05032404959926932094125265068, 10.69259498468142845607361922200, 11.10712776078351919528325205285

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