Properties

Label 8-570e4-1.1-c5e4-0-6
Degree $8$
Conductor $105560010000$
Sign $1$
Analytic cond. $6.98460\times 10^{7}$
Root an. cond. $9.56131$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·2-s + 36·3-s + 160·4-s + 100·5-s − 576·6-s − 26·7-s − 1.28e3·8-s + 810·9-s − 1.60e3·10-s − 336·11-s + 5.76e3·12-s − 734·13-s + 416·14-s + 3.60e3·15-s + 8.96e3·16-s + 480·17-s − 1.29e4·18-s − 1.44e3·19-s + 1.60e4·20-s − 936·21-s + 5.37e3·22-s − 1.96e3·23-s − 4.60e4·24-s + 6.25e3·25-s + 1.17e4·26-s + 1.45e4·27-s − 4.16e3·28-s + ⋯
L(s)  = 1  − 2.82·2-s + 2.30·3-s + 5·4-s + 1.78·5-s − 6.53·6-s − 0.200·7-s − 7.07·8-s + 10/3·9-s − 5.05·10-s − 0.837·11-s + 11.5·12-s − 1.20·13-s + 0.567·14-s + 4.13·15-s + 35/4·16-s + 0.402·17-s − 9.42·18-s − 0.917·19-s + 8.94·20-s − 0.463·21-s + 2.36·22-s − 0.773·23-s − 16.3·24-s + 2·25-s + 3.40·26-s + 3.84·27-s − 1.00·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(6.98460\times 10^{7}\)
Root analytic conductor: \(9.56131\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 + p^{2} T )^{4} \)
3$C_1$ \( ( 1 - p^{2} T )^{4} \)
5$C_1$ \( ( 1 - p^{2} T )^{4} \)
19$C_1$ \( ( 1 + p^{2} T )^{4} \)
good7$C_2 \wr S_4$ \( 1 + 26 T + 50968 T^{2} + 261566 T^{3} + 1131090054 T^{4} + 261566 p^{5} T^{5} + 50968 p^{10} T^{6} + 26 p^{15} T^{7} + p^{20} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 336 T + 377460 T^{2} + 175893684 T^{3} + 69532824974 T^{4} + 175893684 p^{5} T^{5} + 377460 p^{10} T^{6} + 336 p^{15} T^{7} + p^{20} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 734 T + 1500420 T^{2} + 801626734 T^{3} + 64426535182 p T^{4} + 801626734 p^{5} T^{5} + 1500420 p^{10} T^{6} + 734 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 480 T + 2392596 T^{2} - 143630320 p T^{3} + 4082707213942 T^{4} - 143630320 p^{6} T^{5} + 2392596 p^{10} T^{6} - 480 p^{15} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 1962 T + 13953280 T^{2} + 15533465322 T^{3} + 104383666432862 T^{4} + 15533465322 p^{5} T^{5} + 13953280 p^{10} T^{6} + 1962 p^{15} T^{7} + p^{20} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 8720 T + 36158772 T^{2} + 14846124876 T^{3} - 173287946533034 T^{4} + 14846124876 p^{5} T^{5} + 36158772 p^{10} T^{6} + 8720 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 240 T + 55153236 T^{2} + 62608676128 T^{3} + 1901305795022550 T^{4} + 62608676128 p^{5} T^{5} + 55153236 p^{10} T^{6} + 240 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 14626 T + 167666956 T^{2} + 739397326386 T^{3} + 6071410879748886 T^{4} + 739397326386 p^{5} T^{5} + 167666956 p^{10} T^{6} + 14626 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 11092 T + 453374004 T^{2} + 3610413442136 T^{3} + 78268684962183206 T^{4} + 3610413442136 p^{5} T^{5} + 453374004 p^{10} T^{6} + 11092 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 16778 T + 442445048 T^{2} + 5868769961134 T^{3} + 91671081225641238 T^{4} + 5868769961134 p^{5} T^{5} + 442445048 p^{10} T^{6} + 16778 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 34810 T + 714328960 T^{2} + 6800749361322 T^{3} + 77413122096422462 T^{4} + 6800749361322 p^{5} T^{5} + 714328960 p^{10} T^{6} + 34810 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 18186 T + 333416220 T^{2} + 5558506356374 T^{3} + 274963800499499782 T^{4} + 5558506356374 p^{5} T^{5} + 333416220 p^{10} T^{6} + 18186 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 38700 T + 479049564 T^{2} + 8583676522316 T^{3} + 288790694624222710 T^{4} + 8583676522316 p^{5} T^{5} + 479049564 p^{10} T^{6} + 38700 p^{15} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 12080 T + 867941948 T^{2} - 43882474121088 T^{3} - 506183550735628874 T^{4} - 43882474121088 p^{5} T^{5} + 867941948 p^{10} T^{6} + 12080 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 84216 T + 3532049612 T^{2} + 35688008235832 T^{3} - 487439309207630602 T^{4} + 35688008235832 p^{5} T^{5} + 3532049612 p^{10} T^{6} + 84216 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 3592 T + 4955236092 T^{2} + 16975042268152 T^{3} + 11495368453717738214 T^{4} + 16975042268152 p^{5} T^{5} + 4955236092 p^{10} T^{6} - 3592 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 41180 T + 7897434916 T^{2} + 224992823711172 T^{3} + 23857078563550088790 T^{4} + 224992823711172 p^{5} T^{5} + 7897434916 p^{10} T^{6} + 41180 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 15272 T + 8461983548 T^{2} - 51551563637256 T^{3} + 33269726048865355078 T^{4} - 51551563637256 p^{5} T^{5} + 8461983548 p^{10} T^{6} - 15272 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 133106 T + 14754043384 T^{2} - 962317274177322 T^{3} + 69516755749122438206 T^{4} - 962317274177322 p^{5} T^{5} + 14754043384 p^{10} T^{6} - 133106 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 133704 T + 24761236868 T^{2} - 2108891159887868 T^{3} + \)\(21\!\cdots\!30\)\( T^{4} - 2108891159887868 p^{5} T^{5} + 24761236868 p^{10} T^{6} - 133704 p^{15} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 161594 T + 32960947988 T^{2} + 3460060000436394 T^{3} + \)\(40\!\cdots\!34\)\( T^{4} + 3460060000436394 p^{5} T^{5} + 32960947988 p^{10} T^{6} + 161594 p^{15} T^{7} + p^{20} T^{8} \)
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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

−7.74388399986924012346836179742, −7.25591382508933456510577241061, −7.19336307486245037741133375246, −6.82936512271195859906760333326, −6.67656854403494001112881598129, −6.36243462885880693681747028910, −6.18654333247477747966192351181, −5.86440000706541614093148610387, −5.84496618749708180461558501601, −5.03644006949666833169510580551, −4.90527509042990215737134279160, −4.80438442177955876823002653438, −4.76189330385257354244032675735, −3.60578913907780868650347355024, −3.51770314775081203669057572685, −3.50026526732509817351450094677, −3.31940564796840064290209000180, −2.59148080287552847768976901930, −2.47614842752999316631014545658, −2.41885927683878113066361511007, −2.17943336712367985216800337504, −1.63191959663384393774799366648, −1.44728942154477083872763478525, −1.40510127894702940549578653460, −1.37653778158877842622807619696, 0, 0, 0, 0, 1.37653778158877842622807619696, 1.40510127894702940549578653460, 1.44728942154477083872763478525, 1.63191959663384393774799366648, 2.17943336712367985216800337504, 2.41885927683878113066361511007, 2.47614842752999316631014545658, 2.59148080287552847768976901930, 3.31940564796840064290209000180, 3.50026526732509817351450094677, 3.51770314775081203669057572685, 3.60578913907780868650347355024, 4.76189330385257354244032675735, 4.80438442177955876823002653438, 4.90527509042990215737134279160, 5.03644006949666833169510580551, 5.84496618749708180461558501601, 5.86440000706541614093148610387, 6.18654333247477747966192351181, 6.36243462885880693681747028910, 6.67656854403494001112881598129, 6.82936512271195859906760333326, 7.19336307486245037741133375246, 7.25591382508933456510577241061, 7.74388399986924012346836179742

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