Properties

Label 4-1568e2-1.1-c2e2-0-7
Degree $4$
Conductor $2458624$
Sign $1$
Analytic cond. $1825.41$
Root an. cond. $6.53642$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 17·9-s + 48·13-s + 2·17-s − 47·25-s + 48·29-s − 98·37-s − 96·41-s + 34·45-s − 50·53-s − 2·61-s + 96·65-s + 190·73-s + 208·81-s + 4·85-s + 190·89-s + 288·97-s + 146·101-s − 142·109-s + 192·113-s + 816·117-s − 47·121-s − 146·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 2/5·5-s + 17/9·9-s + 3.69·13-s + 2/17·17-s − 1.87·25-s + 1.65·29-s − 2.64·37-s − 2.34·41-s + 0.755·45-s − 0.943·53-s − 0.0327·61-s + 1.47·65-s + 2.60·73-s + 2.56·81-s + 4/85·85-s + 2.13·89-s + 2.96·97-s + 1.44·101-s − 1.30·109-s + 1.69·113-s + 6.97·117-s − 0.388·121-s − 1.16·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2458624\)    =    \(2^{10} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1825.41\)
Root analytic conductor: \(6.53642\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2458624,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.420543173\)
\(L(\frac12)\) \(\approx\) \(5.420543173\)
\(L(2)\) 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)$
bad2 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 17 T^{2} + p^{4} T^{4} \)
5$C_2$ \( ( 1 - T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 + 47 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 - T + p^{2} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 673 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - 1009 T^{2} + p^{4} T^{4} \)
29$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 241 T^{2} + p^{4} T^{4} \)
37$C_2$ \( ( 1 + 49 T + p^{2} T^{2} )^{2} \)
41$C_2$ \( ( 1 + 48 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 3122 T^{2} + p^{4} T^{4} \)
47$C_2^2$ \( 1 - 1393 T^{2} + p^{4} T^{4} \)
53$C_2$ \( ( 1 + 25 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 6673 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 4753 T^{2} + p^{4} T^{4} \)
71$C_2^2$ \( 1 - 866 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 95 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 10801 T^{2} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 8594 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 95 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 144 T + p^{2} T^{2} )^{2} \)
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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

−9.406719406852110997081303780287, −8.988989986484869830239968770130, −8.540410323146462936999189577183, −8.468752889184557884717042750151, −7.75551583448930248518713441600, −7.64319149725720321384762697754, −6.79243482474875009519434986261, −6.51857873185918851434755613298, −6.36812905603996545269053181644, −5.96497737865809028549843827158, −5.16670520761857642507571168290, −5.04606263594184858383930194039, −4.29688048091104946226541468456, −3.81288014354592734396648854040, −3.41876164858198585331449951423, −3.41245638391770114190056805296, −1.93309475750444386976493939301, −1.82915816527752535301155794634, −1.27849246210382294034575121558, −0.69621390256156682547718037803, 0.69621390256156682547718037803, 1.27849246210382294034575121558, 1.82915816527752535301155794634, 1.93309475750444386976493939301, 3.41245638391770114190056805296, 3.41876164858198585331449951423, 3.81288014354592734396648854040, 4.29688048091104946226541468456, 5.04606263594184858383930194039, 5.16670520761857642507571168290, 5.96497737865809028549843827158, 6.36812905603996545269053181644, 6.51857873185918851434755613298, 6.79243482474875009519434986261, 7.64319149725720321384762697754, 7.75551583448930248518713441600, 8.468752889184557884717042750151, 8.540410323146462936999189577183, 8.988989986484869830239968770130, 9.406719406852110997081303780287

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