Properties

Label 4-1148e2-1.1-c1e2-0-4
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 − 4·7-s − 3·8-s + 2·9-s − 2·11-s − 4·14-s − 16-s + 2·18-s − 2·22-s + 12·23-s + 2·25-s + 4·28-s − 14·29-s + 5·32-s − 2·36-s + 4·37-s − 8·43-s + 2·44-s + 12·46-s + 9·49-s + 2·50-s + 6·53-s + 12·56-s − 14·58-s − 8·63-s + 7·64-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.51·7-s − 1.06·8-s + 2/3·9-s − 0.603·11-s − 1.06·14-s − 1/4·16-s + 0.471·18-s − 0.426·22-s + 2.50·23-s + 2/5·25-s + 0.755·28-s − 2.59·29-s + 0.883·32-s − 1/3·36-s + 0.657·37-s − 1.21·43-s + 0.301·44-s + 1.76·46-s + 9/7·49-s + 0.282·50-s + 0.824·53-s + 1.60·56-s − 1.83·58-s − 1.00·63-s + 7/8·64-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$C_2$ \( 1 + 4 T + p T^{2} \)
41$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
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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

−7.56733619312816819531740305261, −7.20421653497416341972870084569, −6.85552111025605262181450181804, −6.51007540812857910327410929642, −5.75706051397583714929402063481, −5.60862750508571607983732563032, −5.12874623491498510803274444388, −4.55793246810234766003248193565, −4.15004975073289992247147290174, −3.53748492184856308530402080877, −3.14323039296420278130805486479, −2.84497367648824844588446125252, −1.96342256826295179045814750714, −0.933505167233924177873085894208, 0, 0.933505167233924177873085894208, 1.96342256826295179045814750714, 2.84497367648824844588446125252, 3.14323039296420278130805486479, 3.53748492184856308530402080877, 4.15004975073289992247147290174, 4.55793246810234766003248193565, 5.12874623491498510803274444388, 5.60862750508571607983732563032, 5.75706051397583714929402063481, 6.51007540812857910327410929642, 6.85552111025605262181450181804, 7.20421653497416341972870084569, 7.56733619312816819531740305261

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