Properties

Label 4-1620e2-1.1-c3e2-0-7
Degree $4$
Conductor $2624400$
Sign $1$
Analytic cond. $9136.12$
Root an. cond. $9.77666$
Motivic weight $3$
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  + 5·5-s + 16·7-s − 60·11-s − 86·13-s − 36·17-s + 88·19-s + 48·23-s − 186·29-s − 176·31-s + 80·35-s + 508·37-s + 186·41-s + 100·43-s + 168·47-s + 343·49-s + 996·53-s − 300·55-s − 252·59-s + 58·61-s − 430·65-s + 1.03e3·67-s − 336·71-s + 1.01e3·73-s − 960·77-s − 272·79-s + 948·83-s − 180·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.863·7-s − 1.64·11-s − 1.83·13-s − 0.513·17-s + 1.06·19-s + 0.435·23-s − 1.19·29-s − 1.01·31-s + 0.386·35-s + 2.25·37-s + 0.708·41-s + 0.354·43-s + 0.521·47-s + 49-s + 2.58·53-s − 0.735·55-s − 0.556·59-s + 0.121·61-s − 0.820·65-s + 1.88·67-s − 0.561·71-s + 1.62·73-s − 1.42·77-s − 0.387·79-s + 1.25·83-s − 0.229·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2624400\)    =    \(2^{4} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(9136.12\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2624400,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.657512447\)
\(L(\frac12)\) \(\approx\) \(3.657512447\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 - p T + p^{2} T^{2} \)
good7$C_2^2$ \( 1 - 16 T - 87 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 60 T + 2269 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2^2$ \( 1 + 86 T + 5199 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2$ \( ( 1 + 18 T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 44 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 48 T - 9863 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 + 186 T + 10207 T^{2} + 186 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2^2$ \( 1 + 176 T + 1185 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2$ \( ( 1 - 254 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 186 T - 34325 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 - 100 T - 69507 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 168 T - 75599 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2$ \( ( 1 - 498 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 252 T - 141875 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 58 T - 223617 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 1036 T + 772533 T^{2} - 1036 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 168 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 506 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 + 272 T - 419055 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2^2$ \( 1 - 948 T + 326917 T^{2} - 948 p^{3} T^{3} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 1014 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 766 T - 325917 T^{2} - 766 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.276534331209755032247706263207, −9.055100537876502153046050223847, −8.281281775387026206272623789040, −7.904356345376900398919844958317, −7.55170625375127014940184902967, −7.50361661724204225904122205033, −6.95096419176899790254904630779, −6.46858962337563083240182925232, −5.61750741845705150279946543302, −5.59538125781267118791102383593, −5.05589447741276233784417044508, −4.99201408668383777456252213252, −4.08872410076742705348402316441, −4.01826878880991430906009923769, −2.91650645943529010021570550827, −2.72086869462026572569919216896, −2.08298483898253741426287811653, −1.98209275957300328714694989695, −0.72499512046531376205107894599, −0.58627047519109592959446195252, 0.58627047519109592959446195252, 0.72499512046531376205107894599, 1.98209275957300328714694989695, 2.08298483898253741426287811653, 2.72086869462026572569919216896, 2.91650645943529010021570550827, 4.01826878880991430906009923769, 4.08872410076742705348402316441, 4.99201408668383777456252213252, 5.05589447741276233784417044508, 5.59538125781267118791102383593, 5.61750741845705150279946543302, 6.46858962337563083240182925232, 6.95096419176899790254904630779, 7.50361661724204225904122205033, 7.55170625375127014940184902967, 7.904356345376900398919844958317, 8.281281775387026206272623789040, 9.055100537876502153046050223847, 9.276534331209755032247706263207

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