Properties

Label 4-5024-1.1-c1e2-0-2
Degree $4$
Conductor $5024$
Sign $-1$
Analytic cond. $0.320334$
Root an. cond. $0.752317$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 7·5-s − 8-s − 9-s + 7·10-s − 4·13-s + 16-s − 2·17-s + 18-s − 7·20-s + 27·25-s + 4·26-s − 4·29-s − 32-s + 2·34-s − 36-s − 4·37-s + 7·40-s − 41-s + 7·45-s + 4·49-s − 27·50-s − 4·52-s + 15·53-s + 4·58-s − 18·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 3.13·5-s − 0.353·8-s − 1/3·9-s + 2.21·10-s − 1.10·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 1.56·20-s + 27/5·25-s + 0.784·26-s − 0.742·29-s − 0.176·32-s + 0.342·34-s − 1/6·36-s − 0.657·37-s + 1.10·40-s − 0.156·41-s + 1.04·45-s + 4/7·49-s − 3.81·50-s − 0.554·52-s + 2.06·53-s + 0.525·58-s − 2.30·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5024\)    =    \(2^{5} \cdot 157\)
Sign: $-1$
Analytic conductor: \(0.320334\)
Root analytic conductor: \(0.752317\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 5024,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
157$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 7 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 21 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 63 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 67 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−11.90056521192628329486620605524, −11.43394942290669023543484205646, −10.91683083415851330908708472091, −10.41521183911198892345090894769, −9.421301358107801462771809309008, −8.747019283078732223949061314581, −8.248398033495562096436455073257, −7.69235284793502487079140479491, −7.27646930234211369073790847492, −6.77482546138016476951023540519, −5.38722891524652456492225142043, −4.39262417042128907042000513978, −3.83390921970190067244943689230, −2.86465139544535780750505633003, 0, 2.86465139544535780750505633003, 3.83390921970190067244943689230, 4.39262417042128907042000513978, 5.38722891524652456492225142043, 6.77482546138016476951023540519, 7.27646930234211369073790847492, 7.69235284793502487079140479491, 8.248398033495562096436455073257, 8.747019283078732223949061314581, 9.421301358107801462771809309008, 10.41521183911198892345090894769, 10.91683083415851330908708472091, 11.43394942290669023543484205646, 11.90056521192628329486620605524

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