Properties

Label 4-155e2-1.1-c1e2-0-0
Degree $4$
Conductor $24025$
Sign $1$
Analytic cond. $1.53185$
Root an. cond. $1.11251$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 5·9-s + 3·11-s − 4·16-s − 4·25-s − 10·29-s − 31-s + 3·41-s + 5·45-s + 5·49-s + 3·55-s + 5·59-s − 2·61-s + 71-s − 5·79-s − 4·80-s + 16·81-s + 10·89-s + 15·99-s − 4·101-s − 10·109-s − 9·121-s − 9·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.447·5-s + 5/3·9-s + 0.904·11-s − 16-s − 4/5·25-s − 1.85·29-s − 0.179·31-s + 0.468·41-s + 0.745·45-s + 5/7·49-s + 0.404·55-s + 0.650·59-s − 0.256·61-s + 0.118·71-s − 0.562·79-s − 0.447·80-s + 16/9·81-s + 1.05·89-s + 1.50·99-s − 0.398·101-s − 0.957·109-s − 0.818·121-s − 0.804·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(24025\)    =    \(5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(1.53185\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 24025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.428053298\)
\(L(\frac12)\) \(\approx\) \(1.428053298\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
31$C_2$ \( 1 + T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \)
13$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 95 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 95 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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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

−10.70917705909503306583588671539, −10.05458370108966505514733389242, −9.621828832141109284258005784986, −9.239371053510812335337630973328, −8.776950382982104263607468888499, −7.83605235154742652087022721420, −7.34589008364915840446400319821, −6.89542555928777320395612330756, −6.29973140469120430294973959024, −5.64025634937085876476102581393, −4.84189736523501903395160367137, −4.09045033490687232695717305489, −3.73687090201400686590050399962, −2.29562056225529276258388423470, −1.51635649866021460384122379637, 1.51635649866021460384122379637, 2.29562056225529276258388423470, 3.73687090201400686590050399962, 4.09045033490687232695717305489, 4.84189736523501903395160367137, 5.64025634937085876476102581393, 6.29973140469120430294973959024, 6.89542555928777320395612330756, 7.34589008364915840446400319821, 7.83605235154742652087022721420, 8.776950382982104263607468888499, 9.239371053510812335337630973328, 9.621828832141109284258005784986, 10.05458370108966505514733389242, 10.70917705909503306583588671539

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