Properties

Label 4-845e2-1.1-c1e2-0-12
Degree $4$
Conductor $714025$
Sign $1$
Analytic cond. $45.5268$
Root an. cond. $2.59756$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4-s − 2·5-s − 4·7-s − 4·9-s − 4·10-s − 4·11-s − 8·14-s + 16-s − 4·17-s − 8·18-s − 4·19-s − 2·20-s − 8·22-s + 3·25-s − 4·28-s − 12·31-s − 2·32-s − 8·34-s + 8·35-s − 4·36-s − 8·38-s + 12·41-s − 8·43-s − 4·44-s + 8·45-s + 4·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s − 0.894·5-s − 1.51·7-s − 4/3·9-s − 1.26·10-s − 1.20·11-s − 2.13·14-s + 1/4·16-s − 0.970·17-s − 1.88·18-s − 0.917·19-s − 0.447·20-s − 1.70·22-s + 3/5·25-s − 0.755·28-s − 2.15·31-s − 0.353·32-s − 1.37·34-s + 1.35·35-s − 2/3·36-s − 1.29·38-s + 1.87·41-s − 1.21·43-s − 0.603·44-s + 1.19·45-s + 0.583·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 714025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(714025\)    =    \(5^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(45.5268\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 714025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad5$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$C_4$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.15767672223446692403396422962, −9.367919148790405954684299632370, −9.026942848136500443879317012396, −8.964072752604296114631071938061, −8.173790422426284434131822630557, −7.64515130132572478231498067471, −7.58237203867389184223307986446, −6.73744656346224362208410858969, −6.36515192198646858037623592060, −5.97031552585204471714607698511, −5.46396933825867778629999014387, −5.09776968870667565798324370254, −4.38993345435487524521668471709, −4.25183294641275271894156570825, −3.33790417930147427701901169465, −3.30964100219849301181980737041, −2.69989485395194313129565717715, −1.96652415285945947577038679902, 0, 0, 1.96652415285945947577038679902, 2.69989485395194313129565717715, 3.30964100219849301181980737041, 3.33790417930147427701901169465, 4.25183294641275271894156570825, 4.38993345435487524521668471709, 5.09776968870667565798324370254, 5.46396933825867778629999014387, 5.97031552585204471714607698511, 6.36515192198646858037623592060, 6.73744656346224362208410858969, 7.58237203867389184223307986446, 7.64515130132572478231498067471, 8.173790422426284434131822630557, 8.964072752604296114631071938061, 9.026942848136500443879317012396, 9.367919148790405954684299632370, 10.15767672223446692403396422962

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