Properties

Label 4-1148e2-1.1-c1e2-0-0
Degree $4$
Conductor $1317904$
Sign $-1$
Analytic cond. $84.0307$
Root an. cond. $3.02767$
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 − 4·5-s − 3·8-s − 6·9-s − 4·10-s − 8·13-s − 16-s − 8·17-s − 6·18-s + 4·20-s + 6·25-s − 8·26-s + 4·29-s + 5·32-s − 8·34-s + 6·36-s + 4·37-s + 12·40-s + 6·41-s + 24·45-s − 49-s + 6·50-s + 8·52-s + 4·53-s + 4·58-s + 4·61-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.06·8-s − 2·9-s − 1.26·10-s − 2.21·13-s − 1/4·16-s − 1.94·17-s − 1.41·18-s + 0.894·20-s + 6/5·25-s − 1.56·26-s + 0.742·29-s + 0.883·32-s − 1.37·34-s + 36-s + 0.657·37-s + 1.89·40-s + 0.937·41-s + 3.57·45-s − 1/7·49-s + 0.848·50-s + 1.10·52-s + 0.549·53-s + 0.525·58-s + 0.512·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1317904\)    =    \(2^{4} \cdot 7^{2} \cdot 41^{2}\)
Sign: $-1$
Analytic conductor: \(84.0307\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1317904} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1317904,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
41$C_2$ \( 1 - 6 T + p T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
67$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−7.78350890135684556717209651799, −7.44917321683684526309857142748, −6.83968101590508539856361241314, −6.43317252490646382128647321188, −5.96492579449815088704891244330, −5.32927734537435592048663519374, −4.97175437942210426498225906306, −4.63904532199712964364097974899, −4.07829807677930754449082312817, −3.83540030579287121471413477871, −3.02480657122313342353826528614, −2.62889149594274994901562173478, −2.35684139074715943914617651701, −0.49761928289442785673056631833, 0, 0.49761928289442785673056631833, 2.35684139074715943914617651701, 2.62889149594274994901562173478, 3.02480657122313342353826528614, 3.83540030579287121471413477871, 4.07829807677930754449082312817, 4.63904532199712964364097974899, 4.97175437942210426498225906306, 5.32927734537435592048663519374, 5.96492579449815088704891244330, 6.43317252490646382128647321188, 6.83968101590508539856361241314, 7.44917321683684526309857142748, 7.78350890135684556717209651799

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