Properties

Label 8-403e4-1.1-c1e4-0-0
Degree $8$
Conductor $26376683281$
Sign $1$
Analytic cond. $107.233$
Root an. cond. $1.79387$
Motivic weight $1$
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  − 2·2-s + 6·3-s + 2·4-s − 2·5-s − 12·6-s − 10·7-s − 2·8-s + 16·9-s + 4·10-s + 6·11-s + 12·12-s + 4·13-s + 20·14-s − 12·15-s + 3·16-s − 32·18-s + 6·19-s − 4·20-s − 60·21-s − 12·22-s − 4·23-s − 12·24-s + 2·25-s − 8·26-s + 24·27-s − 20·28-s + 24·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 3.46·3-s + 4-s − 0.894·5-s − 4.89·6-s − 3.77·7-s − 0.707·8-s + 16/3·9-s + 1.26·10-s + 1.80·11-s + 3.46·12-s + 1.10·13-s + 5.34·14-s − 3.09·15-s + 3/4·16-s − 7.54·18-s + 1.37·19-s − 0.894·20-s − 13.0·21-s − 2.55·22-s − 0.834·23-s − 2.44·24-s + 2/5·25-s − 1.56·26-s + 4.61·27-s − 3.77·28-s + 4.38·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(13^{4} \cdot 31^{4}\)
Sign: $1$
Analytic conductor: \(107.233\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 13^{4} \cdot 31^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.8679770374\)
\(L(\frac12)\) \(\approx\) \(0.8679770374\)
\(L(\frac{3}{2})\) 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)$
bad13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
good2$D_4\times C_2$ \( 1 + p T + p T^{2} + p T^{3} + T^{4} + p^{2} T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
3$D_4\times C_2$ \( 1 - 2 p T + 20 T^{2} - 16 p T^{3} + 91 T^{4} - 16 p^{2} T^{5} + 20 p^{2} T^{6} - 2 p^{4} T^{7} + p^{4} T^{8} \)
5$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
7$C_2$$\times$$C_2^2$ \( ( 1 + 5 T + p T^{2} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} ) \)
11$D_4\times C_2$ \( 1 - 6 T + 9 T^{2} + 6 p T^{3} - 376 T^{4} + 6 p^{2} T^{5} + 9 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 6 T + 45 T^{2} - 126 T^{3} + 704 T^{4} - 126 p T^{5} + 45 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2$$\times$$C_2^2$ \( ( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
37$C_2^3$ \( 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 288 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 + 20 T + 4 p T^{2} + 668 T^{3} + 2335 T^{4} + 668 p T^{5} + 4 p^{3} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 2 T - 56 T^{2} + 52 T^{3} + 1579 T^{4} + 52 p T^{5} - 56 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 20 T + 200 T^{2} - 1460 T^{3} + 9982 T^{4} - 1460 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 30 T + 9 p T^{2} + 5310 T^{3} + 44420 T^{4} + 5310 p T^{5} + 9 p^{3} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 22 T + 122 T^{2} - 1232 T^{3} - 20033 T^{4} - 1232 p T^{5} + 122 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 182 T^{2} + 15291 T^{4} - 182 p^{2} T^{6} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 8 T + 137 T^{2} - 1224 T^{3} + 11492 T^{4} - 1224 p T^{5} + 137 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 16 T + 65 T^{2} + 1016 T^{3} - 15428 T^{4} + 1016 p T^{5} + 65 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 + 6 T + 90 T^{2} - 564 T^{3} - 97 T^{4} - 564 p T^{5} + 90 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 36 T + 694 T^{2} + 9432 T^{3} + 96531 T^{4} + 9432 p T^{5} + 694 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 38 T + 650 T^{2} - 6944 T^{3} + 62479 T^{4} - 6944 p T^{5} + 650 p^{2} T^{6} - 38 p^{3} T^{7} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 + 16 T + 128 T^{2} + 1840 T^{3} + 25774 T^{4} + 1840 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 20 T + 200 T^{2} + 2460 T^{3} + 29582 T^{4} + 2460 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
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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

−8.246446387793109322512084710025, −8.047021162570699761605908940292, −7.85954894024810863011227386949, −7.53098116406217798335880917742, −7.40530936352630435532067395391, −6.85263212451028725426053849051, −6.82416774919081438415016428667, −6.64418530187372659717682553437, −6.54872542306820916703688463361, −5.99432635456497654457165043484, −5.82633496234183685250966494809, −5.57967274426101864254325791023, −5.03275369666209040857112517831, −4.42577085590612606276828839841, −4.00017145345799191075595290064, −3.77324730109972719353251727106, −3.67699327764048254848188424176, −3.27827021633133061898034685566, −3.18876973713278519387747870444, −3.10979634619315297680477703241, −2.79772557084370523612026589067, −2.30800384652830737695782049785, −1.57253453361196705947115387063, −1.49668887256583417189692956503, −0.35109151455652498844391362959, 0.35109151455652498844391362959, 1.49668887256583417189692956503, 1.57253453361196705947115387063, 2.30800384652830737695782049785, 2.79772557084370523612026589067, 3.10979634619315297680477703241, 3.18876973713278519387747870444, 3.27827021633133061898034685566, 3.67699327764048254848188424176, 3.77324730109972719353251727106, 4.00017145345799191075595290064, 4.42577085590612606276828839841, 5.03275369666209040857112517831, 5.57967274426101864254325791023, 5.82633496234183685250966494809, 5.99432635456497654457165043484, 6.54872542306820916703688463361, 6.64418530187372659717682553437, 6.82416774919081438415016428667, 6.85263212451028725426053849051, 7.40530936352630435532067395391, 7.53098116406217798335880917742, 7.85954894024810863011227386949, 8.047021162570699761605908940292, 8.246446387793109322512084710025

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