Properties

Label 8-405e4-1.1-c1e4-0-12
Degree $8$
Conductor $26904200625$
Sign $1$
Analytic cond. $109.377$
Root an. cond. $1.79831$
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  + 3·2-s + 4·4-s + 3·5-s + 12·7-s + 3·8-s + 9·10-s − 3·11-s + 3·13-s + 36·14-s + 16-s − 10·19-s + 12·20-s − 9·22-s + 6·23-s + 5·25-s + 9·26-s + 48·28-s − 7·31-s + 6·32-s + 36·35-s − 30·38-s + 9·40-s + 3·41-s + 12·43-s − 12·44-s + 18·46-s − 18·47-s + ⋯
L(s)  = 1  + 2.12·2-s + 2·4-s + 1.34·5-s + 4.53·7-s + 1.06·8-s + 2.84·10-s − 0.904·11-s + 0.832·13-s + 9.62·14-s + 1/4·16-s − 2.29·19-s + 2.68·20-s − 1.91·22-s + 1.25·23-s + 25-s + 1.76·26-s + 9.07·28-s − 1.25·31-s + 1.06·32-s + 6.08·35-s − 4.86·38-s + 1.42·40-s + 0.468·41-s + 1.82·43-s − 1.80·44-s + 2.65·46-s − 2.62·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(109.377\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{16} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(16.67354870\)
\(L(\frac12)\) \(\approx\) \(16.67354870\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
good2$D_4\times C_2$ \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + 5 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \)
11$D_4\times C_2$ \( 1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 18 p T^{5} - 7 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 - 3 T + 5 T^{2} - 6 T^{3} - 126 T^{4} - 6 p T^{5} + 5 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 61 T^{2} + 1500 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 6 T + 50 T^{2} - 228 T^{3} + 1191 T^{4} - 228 p T^{5} + 50 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^3$ \( 1 - 25 T^{2} - 216 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 + 7 T - 17 T^{2} + 28 T^{3} + 1876 T^{4} + 28 p T^{5} - 17 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 85 T^{2} + 3876 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 3 T - 67 T^{2} + 18 T^{3} + 3726 T^{4} + 18 p T^{5} - 67 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2^2$ \( ( 1 - 6 T + 55 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 18 T + 218 T^{2} + 1980 T^{3} + 14967 T^{4} + 1980 p T^{5} + 218 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 40 T^{2} + 2718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 9 T - 49 T^{2} + 108 T^{3} + 8640 T^{4} + 108 p T^{5} - 49 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 13 T + 13 T^{2} + 442 T^{3} + 10306 T^{4} + 442 p T^{5} + 13 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 6 T + 50 T^{2} + 228 T^{3} - 2241 T^{4} + 228 p T^{5} + 50 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 229 T^{2} + 23100 T^{4} - 229 p^{2} T^{6} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 2 T - 122 T^{2} + 64 T^{3} + 9319 T^{4} + 64 p T^{5} - 122 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 30 T + 530 T^{2} - 6900 T^{3} + 70911 T^{4} - 6900 p T^{5} + 530 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
97$D_4\times C_2$ \( 1 - 30 T + 470 T^{2} - 5100 T^{3} + 48591 T^{4} - 5100 p T^{5} + 470 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
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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

−8.050693119055952814410540334217, −7.87336905242876427018248083427, −7.75948769102084521591867248910, −7.52647539090682367838501504947, −7.10716522883572909767460595048, −6.75458117853901489325220944559, −6.53139223598780537474420776547, −6.10662732087322543181469844625, −6.05808043814431237370364915415, −5.66961564323328366589907725148, −5.44511324009026278280132152031, −5.20753414674607111990970021140, −4.93480076517137273559271474609, −4.82317812270950281888451307808, −4.55470365631052148218914378790, −4.42817926571289451701335508230, −4.14901285483434908517053176987, −3.88344089987601142274305047382, −3.19033413445418992564815279082, −2.97927262985512945510096480213, −2.37701118548688687221684422504, −2.24561715891817798487480735297, −1.80804609096258264259078382532, −1.41886856847959031644662855358, −1.34904551419603019897360392249, 1.34904551419603019897360392249, 1.41886856847959031644662855358, 1.80804609096258264259078382532, 2.24561715891817798487480735297, 2.37701118548688687221684422504, 2.97927262985512945510096480213, 3.19033413445418992564815279082, 3.88344089987601142274305047382, 4.14901285483434908517053176987, 4.42817926571289451701335508230, 4.55470365631052148218914378790, 4.82317812270950281888451307808, 4.93480076517137273559271474609, 5.20753414674607111990970021140, 5.44511324009026278280132152031, 5.66961564323328366589907725148, 6.05808043814431237370364915415, 6.10662732087322543181469844625, 6.53139223598780537474420776547, 6.75458117853901489325220944559, 7.10716522883572909767460595048, 7.52647539090682367838501504947, 7.75948769102084521591867248910, 7.87336905242876427018248083427, 8.050693119055952814410540334217

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