Properties

Label 4-546e2-1.1-c1e2-0-73
Degree $4$
Conductor $298116$
Sign $1$
Analytic cond. $19.0081$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 3·4-s − 6·5-s − 6·6-s − 7-s + 4·8-s + 6·9-s − 12·10-s − 6·11-s − 9·12-s − 5·13-s − 2·14-s + 18·15-s + 5·16-s + 12·18-s + 5·19-s − 18·20-s + 3·21-s − 12·22-s − 12·24-s + 19·25-s − 10·26-s − 9·27-s − 3·28-s − 12·29-s + 36·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.73·3-s + 3/2·4-s − 2.68·5-s − 2.44·6-s − 0.377·7-s + 1.41·8-s + 2·9-s − 3.79·10-s − 1.80·11-s − 2.59·12-s − 1.38·13-s − 0.534·14-s + 4.64·15-s + 5/4·16-s + 2.82·18-s + 1.14·19-s − 4.02·20-s + 0.654·21-s − 2.55·22-s − 2.44·24-s + 19/5·25-s − 1.96·26-s − 1.73·27-s − 0.566·28-s − 2.22·29-s + 6.57·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(298116\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(19.0081\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{546} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
3$C_2$ \( 1 + p T + p T^{2} \)
7$C_2$ \( 1 + T + p T^{2} \)
13$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 18 T + 149 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 12 T + 95 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T + 65 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 166 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
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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

−10.90952866062036696354132516746, −10.51650103364840739666380023426, −9.881803816249373632320994045217, −9.634296246168040498540408555709, −8.560900719365241778792062309395, −7.86224625421357269552712601822, −7.58244845196947454945730049886, −7.50705575535959942262475892890, −6.77259000409162600257752862291, −6.65407486869489676414617714753, −5.55371251653496028374813728760, −5.27733040700533496377133635086, −5.02291177319682035734253706545, −4.59844033316176412965575440965, −3.84841713519663730346958411568, −3.48011723194157082190756536934, −3.02099837662712722904490683369, −1.84525002752373343507338707616, 0, 0, 1.84525002752373343507338707616, 3.02099837662712722904490683369, 3.48011723194157082190756536934, 3.84841713519663730346958411568, 4.59844033316176412965575440965, 5.02291177319682035734253706545, 5.27733040700533496377133635086, 5.55371251653496028374813728760, 6.65407486869489676414617714753, 6.77259000409162600257752862291, 7.50705575535959942262475892890, 7.58244845196947454945730049886, 7.86224625421357269552712601822, 8.560900719365241778792062309395, 9.634296246168040498540408555709, 9.881803816249373632320994045217, 10.51650103364840739666380023426, 10.90952866062036696354132516746

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