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Properties

Label 8-325e4-1.1-c2e4-0-3
Degree $8$
Conductor $11156640625$
Sign $1$
Analytic cond. $6149.97$
Root an. cond. $2.97583$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·3-s − 4-s + 4·6-s − 16·7-s + 16·9-s + 4·11-s − 2·12-s + 26·13-s − 32·14-s + 12·16-s + 12·17-s + 32·18-s + 10·19-s − 32·21-s + 8·22-s − 18·23-s + 52·26-s + 52·27-s + 16·28-s + 2·29-s − 20·31-s + 26·32-s + 8·33-s + 24·34-s − 16·36-s + 68·37-s + ⋯
L(s)  = 1  + 2-s + 2/3·3-s − 1/4·4-s + 2/3·6-s − 2.28·7-s + 16/9·9-s + 4/11·11-s − 1/6·12-s + 2·13-s − 2.28·14-s + 3/4·16-s + 0.705·17-s + 16/9·18-s + 0.526·19-s − 1.52·21-s + 4/11·22-s − 0.782·23-s + 2·26-s + 1.92·27-s + 4/7·28-s + 2/29·29-s − 0.645·31-s + 0.812·32-s + 8/33·33-s + 0.705·34-s − 4/9·36-s + 1.83·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(6149.97\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 13^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.768028683\)
\(L(\frac12)\) \(\approx\) \(4.768028683\)
\(L(2)\) 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)$
bad5 \( 1 \)
13$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
good2$D_4\times C_2$ \( 1 - p T + 5 T^{2} - 3 p^{2} T^{3} + 17 T^{4} - 3 p^{4} T^{5} + 5 p^{4} T^{6} - p^{7} T^{7} + p^{8} T^{8} \)
3$D_4\times C_2$ \( 1 - 2 T - 4 p T^{2} + 4 T^{3} + 139 T^{4} + 4 p^{2} T^{5} - 4 p^{5} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} \)
7$D_4\times C_2$ \( 1 + 16 T + 164 T^{2} + 1236 T^{3} + 8927 T^{4} + 1236 p^{2} T^{5} + 164 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \)
11$D_4\times C_2$ \( 1 - 4 T + 200 T^{2} + 960 T^{3} + 17471 T^{4} + 960 p^{2} T^{5} + 200 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 - 12 T + 413 T^{2} - 4380 T^{3} + 63576 T^{4} - 4380 p^{2} T^{5} + 413 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 - 10 T + 74 T^{2} - 1752 T^{3} - 96625 T^{4} - 1752 p^{2} T^{5} + 74 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 + 18 T + 968 T^{2} + 15480 T^{3} + 516891 T^{4} + 15480 p^{2} T^{5} + 968 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \)
29$D_4\times C_2$ \( 1 - 2 T - 1571 T^{2} + 214 T^{3} + 1769980 T^{4} + 214 p^{2} T^{5} - 1571 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} \)
31$D_4\times C_2$ \( 1 + 20 T + 200 T^{2} + 18300 T^{3} + 1672334 T^{4} + 18300 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \)
37$D_4\times C_2$ \( 1 - 68 T + 41 p T^{2} + 93168 T^{3} - 5925028 T^{4} + 93168 p^{2} T^{5} + 41 p^{5} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} \)
41$D_4\times C_2$ \( 1 - 100 T + 3461 T^{2} + 7272 T^{3} - 4096804 T^{4} + 7272 p^{2} T^{5} + 3461 p^{4} T^{6} - 100 p^{6} T^{7} + p^{8} T^{8} \)
43$C_2^2$ \( ( 1 + 90 T + 4549 T^{2} + 90 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 68 T + 2312 T^{2} - 148716 T^{3} + 9565454 T^{4} - 148716 p^{2} T^{5} + 2312 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 + 64 T + 4455 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 164 T + 6980 T^{2} - 551844 T^{3} - 68910913 T^{4} - 551844 p^{2} T^{5} + 6980 p^{4} T^{6} + 164 p^{6} T^{7} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 + 124 T + 5413 T^{2} + 312604 T^{3} + 28201432 T^{4} + 312604 p^{2} T^{5} + 5413 p^{4} T^{6} + 124 p^{6} T^{7} + p^{8} T^{8} \)
67$D_4\times C_2$ \( 1 + 118 T + 10706 T^{2} + 763488 T^{3} + 51177839 T^{4} + 763488 p^{2} T^{5} + 10706 p^{4} T^{6} + 118 p^{6} T^{7} + p^{8} T^{8} \)
71$D_4\times C_2$ \( 1 + 86 T + 4658 T^{2} + 258336 T^{3} + 1457087 T^{4} + 258336 p^{2} T^{5} + 4658 p^{4} T^{6} + 86 p^{6} T^{7} + p^{8} T^{8} \)
73$D_4\times C_2$ \( 1 + 58 T + 1682 T^{2} + 201144 T^{3} + 20590727 T^{4} + 201144 p^{2} T^{5} + 1682 p^{4} T^{6} + 58 p^{6} T^{7} + p^{8} T^{8} \)
79$D_{4}$ \( ( 1 + 20 T + 7290 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 188 T + 17672 T^{2} - 1935084 T^{3} + 200304482 T^{4} - 1935084 p^{2} T^{5} + 17672 p^{4} T^{6} - 188 p^{6} T^{7} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 + 110 T + 12050 T^{2} + 1194180 T^{3} + 87746159 T^{4} + 1194180 p^{2} T^{5} + 12050 p^{4} T^{6} + 110 p^{6} T^{7} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 + 178 T + 13250 T^{2} + 318828 T^{3} - 39375793 T^{4} + 318828 p^{2} T^{5} + 13250 p^{4} T^{6} + 178 p^{6} T^{7} + p^{8} T^{8} \)
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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

−8.275130431564399386939725183347, −7.74345265141308813004466692216, −7.71898875282326866418493364040, −7.54913128864956785521584585635, −7.24007918227982743996448991467, −6.58735385193861189354131326799, −6.56617578496520108333253429649, −6.53396719100565809309101680877, −6.00087288129670190534517513287, −5.95386660360421662906787096989, −5.80914148037364912617833591871, −5.16373388566569439269222677677, −4.82400216553094199645230349346, −4.48931154238928254744627582919, −4.45337570390845562523542902829, −3.98292316368484198281885215113, −3.82440007241116884110295603572, −3.55585179650034429608164199859, −2.97733917167772075836073643147, −2.95753673417211117740540915106, −2.94020117039647099637247728668, −1.70037229089411101578732805789, −1.51613323412008809140422631427, −1.25113196580713985896642917563, −0.40745456753584583942618976165, 0.40745456753584583942618976165, 1.25113196580713985896642917563, 1.51613323412008809140422631427, 1.70037229089411101578732805789, 2.94020117039647099637247728668, 2.95753673417211117740540915106, 2.97733917167772075836073643147, 3.55585179650034429608164199859, 3.82440007241116884110295603572, 3.98292316368484198281885215113, 4.45337570390845562523542902829, 4.48931154238928254744627582919, 4.82400216553094199645230349346, 5.16373388566569439269222677677, 5.80914148037364912617833591871, 5.95386660360421662906787096989, 6.00087288129670190534517513287, 6.53396719100565809309101680877, 6.56617578496520108333253429649, 6.58735385193861189354131326799, 7.24007918227982743996448991467, 7.54913128864956785521584585635, 7.71898875282326866418493364040, 7.74345265141308813004466692216, 8.275130431564399386939725183347

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