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Properties

Label 8-78e4-1.1-c2e4-0-2
Degree $8$
Conductor $37015056$
Sign $1$
Analytic cond. $20.4041$
Root an. cond. $1.45785$
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 + 2·4-s + 6·5-s + 20·7-s + 4·8-s − 3·9-s + 12·10-s − 36·11-s + 40·14-s + 8·16-s − 12·17-s − 6·18-s + 44·19-s + 12·20-s − 72·22-s + 12·23-s + 18·25-s + 40·28-s − 60·29-s + 88·31-s + 8·32-s − 24·34-s + 120·35-s − 6·36-s − 2·37-s + 88·38-s + 24·40-s + ⋯
L(s)  = 1  + 2-s + 1/2·4-s + 6/5·5-s + 20/7·7-s + 1/2·8-s − 1/3·9-s + 6/5·10-s − 3.27·11-s + 20/7·14-s + 1/2·16-s − 0.705·17-s − 1/3·18-s + 2.31·19-s + 3/5·20-s − 3.27·22-s + 0.521·23-s + 0.719·25-s + 10/7·28-s − 2.06·29-s + 2.83·31-s + 1/4·32-s − 0.705·34-s + 24/7·35-s − 1/6·36-s − 0.0540·37-s + 2.31·38-s + 3/5·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(37015056\)    =    \(2^{4} \cdot 3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(20.4041\)
Root analytic conductor: \(1.45785\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 37015056,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.832268398\)
\(L(\frac12)\) \(\approx\) \(4.832268398\)
\(L(2)\) 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)$
bad2$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - p^{2} T^{2} + p^{4} T^{4} \)
good5$D_4\times C_2$ \( 1 - 6 T + 18 T^{2} - 168 T^{3} + 1559 T^{4} - 168 p^{2} T^{5} + 18 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} \)
7$D_4\times C_2$ \( 1 - 20 T + 296 T^{2} - 408 p T^{3} + 23375 T^{4} - 408 p^{3} T^{5} + 296 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} \)
11$D_4\times C_2$ \( 1 + 36 T + 360 T^{2} - 2616 T^{3} - 77953 T^{4} - 2616 p^{2} T^{5} + 360 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 + 12 T + 197 T^{2} + 1788 T^{3} - 47448 T^{4} + 1788 p^{2} T^{5} + 197 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 - 44 T + 1160 T^{2} - 22440 T^{3} + 421151 T^{4} - 22440 p^{2} T^{5} + 1160 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 - 12 T + 794 T^{2} - 8952 T^{3} + 302067 T^{4} - 8952 p^{2} T^{5} + 794 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \)
29$D_4\times C_2$ \( 1 + 60 T + 1093 T^{2} + 49500 T^{3} + 2550168 T^{4} + 49500 p^{2} T^{5} + 1093 p^{4} T^{6} + 60 p^{6} T^{7} + p^{8} T^{8} \)
31$D_4\times C_2$ \( 1 - 88 T + 3872 T^{2} - 167640 T^{3} + 6366914 T^{4} - 167640 p^{2} T^{5} + 3872 p^{4} T^{6} - 88 p^{6} T^{7} + p^{8} T^{8} \)
37$D_4\times C_2$ \( 1 + 2 T + 197 T^{2} + 34278 T^{3} - 1466728 T^{4} + 34278 p^{2} T^{5} + 197 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} \)
41$D_4\times C_2$ \( 1 + 48 T + 1017 T^{2} - 108828 T^{3} - 5656492 T^{4} - 108828 p^{2} T^{5} + 1017 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \)
43$D_4\times C_2$ \( 1 - 84 T + 4874 T^{2} - 211848 T^{3} + 7290531 T^{4} - 211848 p^{2} T^{5} + 4874 p^{4} T^{6} - 84 p^{6} T^{7} + p^{8} T^{8} \)
47$C_2^2$ \( ( 1 + 60 T + 1800 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 - 30 T + 4391 T^{2} - 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 216 T + 13968 T^{2} + 201144 T^{3} - 56485873 T^{4} + 201144 p^{2} T^{5} + 13968 p^{4} T^{6} - 216 p^{6} T^{7} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 - 126 T + 4897 T^{2} - 445662 T^{3} + 47618004 T^{4} - 445662 p^{2} T^{5} + 4897 p^{4} T^{6} - 126 p^{6} T^{7} + p^{8} T^{8} \)
67$D_4\times C_2$ \( 1 + 124 T + 5960 T^{2} + 23424 T^{3} - 16063201 T^{4} + 23424 p^{2} T^{5} + 5960 p^{4} T^{6} + 124 p^{6} T^{7} + p^{8} T^{8} \)
71$C_2^3$ \( 1 - 84 T + 3528 T^{2} - 98784 T^{3} - 12172273 T^{4} - 98784 p^{2} T^{5} + 3528 p^{4} T^{6} - 84 p^{6} T^{7} + p^{8} T^{8} \)
73$D_4\times C_2$ \( 1 + 178 T + 15842 T^{2} + 1362768 T^{3} + 111813743 T^{4} + 1362768 p^{2} T^{5} + 15842 p^{4} T^{6} + 178 p^{6} T^{7} + p^{8} T^{8} \)
79$D_{4}$ \( ( 1 + 96 T + 13058 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 168 T + 14112 T^{2} - 717864 T^{3} + 29673602 T^{4} - 717864 p^{2} T^{5} + 14112 p^{4} T^{6} - 168 p^{6} T^{7} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 + 54 T + 11754 T^{2} + 1206696 T^{3} + 98199839 T^{4} + 1206696 p^{2} T^{5} + 11754 p^{4} T^{6} + 54 p^{6} T^{7} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 + 2 p T + 45890 T^{2} + 5361984 T^{3} + 686386559 T^{4} + 5361984 p^{2} T^{5} + 45890 p^{4} T^{6} + 2 p^{7} T^{7} + p^{8} T^{8} \)
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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

−10.61895277601827403487264880838, −10.17427490934116220143505482233, −9.896193995118790347644771205747, −9.713391699712137228467811964590, −9.586440370901569575103216749606, −8.645221100323159297636829906342, −8.578708701876584785590095623752, −8.132555249438515830822131769800, −8.012495736319474737216201064691, −7.960563421026368266781007428204, −7.26950290388799985155144269871, −7.00957231493137362623788032701, −6.93855916397456937589110878590, −5.90845196077048521431817502011, −5.56978328898653831021750454337, −5.55821373985764018857230094655, −5.15472722294144304002350835749, −5.05133854277541149603280165391, −4.47170930353064340930595193076, −4.44705157578227835910099073746, −3.38931376753971293508525940402, −2.84553333759888986071862184326, −2.47347525729957033560410882985, −1.97329089849788707063180145632, −1.26231715631372448872146526690, 1.26231715631372448872146526690, 1.97329089849788707063180145632, 2.47347525729957033560410882985, 2.84553333759888986071862184326, 3.38931376753971293508525940402, 4.44705157578227835910099073746, 4.47170930353064340930595193076, 5.05133854277541149603280165391, 5.15472722294144304002350835749, 5.55821373985764018857230094655, 5.56978328898653831021750454337, 5.90845196077048521431817502011, 6.93855916397456937589110878590, 7.00957231493137362623788032701, 7.26950290388799985155144269871, 7.960563421026368266781007428204, 8.012495736319474737216201064691, 8.132555249438515830822131769800, 8.578708701876584785590095623752, 8.645221100323159297636829906342, 9.586440370901569575103216749606, 9.713391699712137228467811964590, 9.896193995118790347644771205747, 10.17427490934116220143505482233, 10.61895277601827403487264880838

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