Properties

Label 4-7920e2-1.1-c1e2-0-17
Degree $4$
Conductor $62726400$
Sign $1$
Analytic cond. $3999.48$
Root an. cond. $7.95245$
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  − 2·5-s + 4·7-s − 2·11-s − 8·19-s + 8·23-s + 3·25-s + 4·29-s − 4·31-s − 8·35-s − 20·37-s + 4·41-s − 12·43-s + 8·47-s + 4·49-s − 4·53-s + 4·55-s + 4·59-s − 4·61-s − 8·67-s + 12·71-s − 16·73-s − 8·77-s − 16·79-s + 20·83-s − 12·89-s + 16·95-s + 4·97-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.51·7-s − 0.603·11-s − 1.83·19-s + 1.66·23-s + 3/5·25-s + 0.742·29-s − 0.718·31-s − 1.35·35-s − 3.28·37-s + 0.624·41-s − 1.82·43-s + 1.16·47-s + 4/7·49-s − 0.549·53-s + 0.539·55-s + 0.520·59-s − 0.512·61-s − 0.977·67-s + 1.42·71-s − 1.87·73-s − 0.911·77-s − 1.80·79-s + 2.19·83-s − 1.27·89-s + 1.64·95-s + 0.406·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(62726400\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(3999.48\)
Root analytic conductor: \(7.95245\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 62726400,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good7$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 12 T + 116 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T + 98 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 154 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 16 T + 204 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−7.50196893551743508311641120842, −7.39552976143730798028710004868, −7.13323736408460665996055281965, −6.72621337704793379006426835851, −6.36338135333306369921578497480, −5.96235729028925883425381193873, −5.29621207190286166254413426758, −5.17668656281360037494360256519, −4.88371939327483745401661212689, −4.60300688022148887735136521727, −4.01254405545711370278356962426, −3.92122848031816831228968099275, −3.20272342454359496617743819658, −3.04524449676728095869573041689, −2.36652994499655995678795707861, −2.02717044804712489857557900404, −1.43487672660619070492223704370, −1.17963226752299774990382012123, 0, 0, 1.17963226752299774990382012123, 1.43487672660619070492223704370, 2.02717044804712489857557900404, 2.36652994499655995678795707861, 3.04524449676728095869573041689, 3.20272342454359496617743819658, 3.92122848031816831228968099275, 4.01254405545711370278356962426, 4.60300688022148887735136521727, 4.88371939327483745401661212689, 5.17668656281360037494360256519, 5.29621207190286166254413426758, 5.96235729028925883425381193873, 6.36338135333306369921578497480, 6.72621337704793379006426835851, 7.13323736408460665996055281965, 7.39552976143730798028710004868, 7.50196893551743508311641120842

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