Properties

Label 4-114e2-1.1-c1e2-0-1
Degree $4$
Conductor $12996$
Sign $1$
Analytic cond. $0.828636$
Root an. cond. $0.954093$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s − 6·5-s − 3·6-s + 2·7-s − 8-s + 6·9-s − 6·10-s + 9·13-s + 2·14-s + 18·15-s − 16-s + 6·17-s + 6·18-s − 8·19-s − 6·21-s + 3·24-s + 19·25-s + 9·26-s − 9·27-s + 6·29-s + 18·30-s + 6·34-s − 12·35-s − 8·38-s − 27·39-s + 6·40-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s − 2.68·5-s − 1.22·6-s + 0.755·7-s − 0.353·8-s + 2·9-s − 1.89·10-s + 2.49·13-s + 0.534·14-s + 4.64·15-s − 1/4·16-s + 1.45·17-s + 1.41·18-s − 1.83·19-s − 1.30·21-s + 0.612·24-s + 19/5·25-s + 1.76·26-s − 1.73·27-s + 1.11·29-s + 3.28·30-s + 1.02·34-s − 2.02·35-s − 1.29·38-s − 4.32·39-s + 0.948·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12996\)    =    \(2^{2} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.828636\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{114} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12996,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5790171652\)
\(L(\frac12)\) \(\approx\) \(0.5790171652\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + p T + p T^{2} \)
19$C_2$ \( 1 + 8 T + p T^{2} \)
good5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 12 T + 103 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−14.42523747140850661344326232074, −12.82243495678826302919942471214, −12.78951134301813824973529789799, −12.20904673297579849637342308621, −11.73925569930681308939593260603, −11.18408077480697387861441470150, −11.16667191009346429714088634197, −10.72444268687988517463598989558, −9.925093853600671066882103237684, −8.643800337145997528836872995992, −8.320897757636082964727315934810, −7.893468515568327552261057284843, −7.18684934220337001163323860335, −6.23502792569374287118793849084, −6.16234758361217626460643157504, −4.96684836277158435161736630940, −4.61031991713176461845988323493, −3.75486857896117614295779222679, −3.67396725005845432295325936298, −0.925932476813854479131803856221, 0.925932476813854479131803856221, 3.67396725005845432295325936298, 3.75486857896117614295779222679, 4.61031991713176461845988323493, 4.96684836277158435161736630940, 6.16234758361217626460643157504, 6.23502792569374287118793849084, 7.18684934220337001163323860335, 7.893468515568327552261057284843, 8.320897757636082964727315934810, 8.643800337145997528836872995992, 9.925093853600671066882103237684, 10.72444268687988517463598989558, 11.16667191009346429714088634197, 11.18408077480697387861441470150, 11.73925569930681308939593260603, 12.20904673297579849637342308621, 12.78951134301813824973529789799, 12.82243495678826302919942471214, 14.42523747140850661344326232074

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