Properties

Degree $4$
Conductor $395136$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 4·5-s − 7-s + 8-s + 9-s − 4·10-s − 8·11-s + 12·13-s − 14-s + 16-s + 18-s − 4·20-s − 8·22-s + 2·25-s + 12·26-s − 28-s + 32-s + 4·35-s + 36-s − 4·40-s − 8·43-s − 8·44-s − 4·45-s + 49-s + 2·50-s + 12·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s − 2.41·11-s + 3.32·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s − 0.894·20-s − 1.70·22-s + 2/5·25-s + 2.35·26-s − 0.188·28-s + 0.176·32-s + 0.676·35-s + 1/6·36-s − 0.632·40-s − 1.21·43-s − 1.20·44-s − 0.596·45-s + 1/7·49-s + 0.282·50-s + 1.66·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(395136\)    =    \(2^{7} \cdot 3^{2} \cdot 7^{3}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{395136} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 395136,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.614088862\)
\(L(\frac12)\) \(\approx\) \(1.614088862\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−8.356918966632091922574869118017, −8.133889765903900854060733905242, −7.907802501954179251622576832429, −7.28965766970175880058840390576, −6.84552480602986516155667793601, −6.23175022246340560839371166306, −5.81340647648643732482215027480, −5.33985014787602837985094112170, −4.73982413848000625426104276677, −4.09458451974208041146506029777, −3.62482887081886485478101246558, −3.47205407811273829747507807541, −2.74481126065377142074711550653, −1.79260226536651551834145289140, −0.62964118914525720642781422582, 0.62964118914525720642781422582, 1.79260226536651551834145289140, 2.74481126065377142074711550653, 3.47205407811273829747507807541, 3.62482887081886485478101246558, 4.09458451974208041146506029777, 4.73982413848000625426104276677, 5.33985014787602837985094112170, 5.81340647648643732482215027480, 6.23175022246340560839371166306, 6.84552480602986516155667793601, 7.28965766970175880058840390576, 7.907802501954179251622576832429, 8.133889765903900854060733905242, 8.356918966632091922574869118017

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