Properties

Label 4-152352-1.1-c1e2-0-3
Degree $4$
Conductor $152352$
Sign $-1$
Analytic cond. $9.71409$
Root an. cond. $1.76543$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 4·5-s − 8-s + 9-s + 4·10-s − 4·13-s + 16-s − 18-s − 4·20-s + 2·25-s + 4·26-s + 12·29-s − 32-s + 36-s + 4·40-s + 20·41-s − 4·45-s − 10·49-s − 2·50-s − 4·52-s + 4·53-s − 12·58-s + 8·61-s + 64-s + 16·65-s − 72-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.353·8-s + 1/3·9-s + 1.26·10-s − 1.10·13-s + 1/4·16-s − 0.235·18-s − 0.894·20-s + 2/5·25-s + 0.784·26-s + 2.22·29-s − 0.176·32-s + 1/6·36-s + 0.632·40-s + 3.12·41-s − 0.596·45-s − 1.42·49-s − 0.282·50-s − 0.554·52-s + 0.549·53-s − 1.57·58-s + 1.02·61-s + 1/8·64-s + 1.98·65-s − 0.117·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(152352\)    =    \(2^{5} \cdot 3^{2} \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(9.71409\)
Root analytic conductor: \(1.76543\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 152352,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
23$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−9.006150174861512066746877136440, −8.259124288211168671734267271865, −8.206972512375746589573867040310, −7.48954786032446015166944686975, −7.43829407920294250781299014536, −6.76482292412091819449305273660, −6.25250569926844252670184221360, −5.51909301075206075300511483684, −4.78570582427944795062347753374, −4.21717016341822456956547144558, −3.92491062937506799522122153513, −2.91631229938310047782795220333, −2.49907678412809416629747058134, −1.15578668685740236008003655593, 0, 1.15578668685740236008003655593, 2.49907678412809416629747058134, 2.91631229938310047782795220333, 3.92491062937506799522122153513, 4.21717016341822456956547144558, 4.78570582427944795062347753374, 5.51909301075206075300511483684, 6.25250569926844252670184221360, 6.76482292412091819449305273660, 7.43829407920294250781299014536, 7.48954786032446015166944686975, 8.206972512375746589573867040310, 8.259124288211168671734267271865, 9.006150174861512066746877136440

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