Properties

Label 8-75e4-1.1-c2e4-0-2
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $17.4415$
Root an. cond. $1.42954$
Motivic weight $2$
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  + 6·4-s − 7·9-s − 5·16-s − 28·19-s + 168·31-s − 42·36-s + 196·49-s − 32·61-s − 180·64-s − 168·76-s − 48·79-s − 32·81-s + 352·109-s − 66·121-s + 1.00e3·124-s + ⋯
L(s)  = 1  + 3/2·4-s − 7/9·9-s − 0.312·16-s − 1.47·19-s + 5.41·31-s − 7/6·36-s + 4·49-s − 0.524·61-s − 2.81·64-s − 2.21·76-s − 0.607·79-s − 0.395·81-s + 3.22·109-s − 0.545·121-s + 8.12·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(17.4415\)
Root analytic conductor: \(1.42954\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.301921419\)
\(L(\frac12)\) \(\approx\) \(2.301921419\)
\(L(2)\) 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)$
bad3$C_2^2$ \( 1 + 7 T^{2} + p^{4} T^{4} \)
5 \( 1 \)
good2$C_2^2$ \( ( 1 - 3 T^{2} + p^{4} T^{4} )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
11$C_2^2$ \( ( 1 + 3 p T^{2} + p^{4} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )^{2}( 1 + 24 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( ( 1 + 567 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 7 T + p^{2} T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 662 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 582 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 42 T + p^{2} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 1138 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 3087 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 1198 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 2262 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 3462 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 2562 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 6953 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 8982 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 9433 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 12 T + p^{2} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 8927 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 97 T + p^{2} T^{2} )^{2}( 1 + 97 T + p^{2} T^{2} )^{2} \)
97$C_2^2$ \( ( 1 - 13918 T^{2} + p^{4} T^{4} )^{2} \)
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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

−10.50981825181986785872294517575, −10.39047092837333457221418145916, −10.15596425832831174120485415096, −9.840655984491762970981446057829, −9.117818490591680144961085547601, −9.050575231139934831003180986404, −8.731207446676533640624337193978, −8.414257734253821439877396532851, −8.028262890006306496922678387450, −7.83918255447649374644294706000, −7.39831720023179465024042714305, −6.85375806299344915528016160779, −6.68691100168710188210643191112, −6.55437423545580095083128906891, −6.14761488436847336722925105830, −5.65703799695834316788849309154, −5.60316886228379095988567536677, −4.51991831460946341657674476768, −4.42812840412616322515225238100, −4.38205075448899025952440477021, −3.24897179604097912545719314892, −2.88055083721362980639495000710, −2.34510386953098010494079647236, −2.22481081588170554204089271884, −0.908229494399085101493476265249, 0.908229494399085101493476265249, 2.22481081588170554204089271884, 2.34510386953098010494079647236, 2.88055083721362980639495000710, 3.24897179604097912545719314892, 4.38205075448899025952440477021, 4.42812840412616322515225238100, 4.51991831460946341657674476768, 5.60316886228379095988567536677, 5.65703799695834316788849309154, 6.14761488436847336722925105830, 6.55437423545580095083128906891, 6.68691100168710188210643191112, 6.85375806299344915528016160779, 7.39831720023179465024042714305, 7.83918255447649374644294706000, 8.028262890006306496922678387450, 8.414257734253821439877396532851, 8.731207446676533640624337193978, 9.050575231139934831003180986404, 9.117818490591680144961085547601, 9.840655984491762970981446057829, 10.15596425832831174120485415096, 10.39047092837333457221418145916, 10.50981825181986785872294517575

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