Properties

Label 4-966e2-1.1-c1e2-0-23
Degree $4$
Conductor $933156$
Sign $1$
Analytic cond. $59.4988$
Root an. cond. $2.77732$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·3-s + 3·4-s + 4·5-s + 4·6-s + 2·7-s + 4·8-s + 3·9-s + 8·10-s + 6·12-s + 4·14-s + 8·15-s + 5·16-s + 6·18-s − 4·19-s + 12·20-s + 4·21-s − 2·23-s + 8·24-s + 2·25-s + 4·27-s + 6·28-s − 4·29-s + 16·30-s + 4·31-s + 6·32-s + 8·35-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.78·5-s + 1.63·6-s + 0.755·7-s + 1.41·8-s + 9-s + 2.52·10-s + 1.73·12-s + 1.06·14-s + 2.06·15-s + 5/4·16-s + 1.41·18-s − 0.917·19-s + 2.68·20-s + 0.872·21-s − 0.417·23-s + 1.63·24-s + 2/5·25-s + 0.769·27-s + 1.13·28-s − 0.742·29-s + 2.92·30-s + 0.718·31-s + 1.06·32-s + 1.35·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(933156\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(59.4988\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 933156,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(12.46211970\)
\(L(\frac12)\) \(\approx\) \(12.46211970\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_4$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 8 T + 14 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−10.26124258674917254550487217205, −9.738192136252859605356064269867, −9.584007996699129219946956093419, −8.906248210020816594514505143451, −8.605673790472952672294072178775, −8.139633493422719129118916760252, −7.52688652905998670386864167423, −7.37588311853937904325025269121, −6.53318796392168469508655334504, −6.38385247650647585026092836777, −5.81434742854424470904257199085, −5.49619527580230857308832085786, −4.87957428627708397509522203188, −4.58425400030105124017947836583, −3.82024851165530888117705794237, −3.63173394894822702705589040126, −2.60336449179103990576532404784, −2.49878402826380940678274663916, −1.67695778472700422371539025888, −1.63244256544766525743959276876, 1.63244256544766525743959276876, 1.67695778472700422371539025888, 2.49878402826380940678274663916, 2.60336449179103990576532404784, 3.63173394894822702705589040126, 3.82024851165530888117705794237, 4.58425400030105124017947836583, 4.87957428627708397509522203188, 5.49619527580230857308832085786, 5.81434742854424470904257199085, 6.38385247650647585026092836777, 6.53318796392168469508655334504, 7.37588311853937904325025269121, 7.52688652905998670386864167423, 8.139633493422719129118916760252, 8.605673790472952672294072178775, 8.906248210020816594514505143451, 9.584007996699129219946956093419, 9.738192136252859605356064269867, 10.26124258674917254550487217205

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