Properties

Label 4-114e2-1.1-c1e2-0-5
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  + 2·2-s − 3·3-s + 3·4-s − 6·6-s + 2·7-s + 4·8-s + 6·9-s − 9·12-s + 4·14-s + 5·16-s + 12·18-s − 8·19-s − 6·21-s − 12·24-s − 2·25-s − 9·27-s + 6·28-s + 18·29-s + 6·32-s + 18·36-s − 16·38-s − 12·42-s + 4·43-s − 15·48-s − 11·49-s − 4·50-s − 18·53-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.73·3-s + 3/2·4-s − 2.44·6-s + 0.755·7-s + 1.41·8-s + 2·9-s − 2.59·12-s + 1.06·14-s + 5/4·16-s + 2.82·18-s − 1.83·19-s − 1.30·21-s − 2.44·24-s − 2/5·25-s − 1.73·27-s + 1.13·28-s + 3.34·29-s + 1.06·32-s + 3·36-s − 2.59·38-s − 1.85·42-s + 0.609·43-s − 2.16·48-s − 1.57·49-s − 0.565·50-s − 2.47·53-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

\(L(1)\) \(\approx\) \(1.516113114\)
\(L(\frac12)\) \(\approx\) \(1.516113114\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
19$C_2$ \( 1 + 8 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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

−13.93865406482178534730604431979, −13.15489901751515059955542535017, −12.61561510748546313828220353693, −12.38098109882178152909384304628, −11.86405326257156272124322486984, −11.43843241121292595298890662752, −10.82738199385273632155144207714, −10.58246001556314574441209511667, −10.11430042684651296706286539675, −9.118119754959359622108277338484, −8.112262278083392832402569165284, −7.76232100624168519610049007081, −6.54564424096198188129094551060, −6.53910247390834776155477036816, −5.99522944989182225075727631750, −5.07148813490284239032473666909, −4.58512735759355047775599264373, −4.36832954540764876987581180628, −2.99063705687272663952860070260, −1.64369780833473190775040709017, 1.64369780833473190775040709017, 2.99063705687272663952860070260, 4.36832954540764876987581180628, 4.58512735759355047775599264373, 5.07148813490284239032473666909, 5.99522944989182225075727631750, 6.53910247390834776155477036816, 6.54564424096198188129094551060, 7.76232100624168519610049007081, 8.112262278083392832402569165284, 9.118119754959359622108277338484, 10.11430042684651296706286539675, 10.58246001556314574441209511667, 10.82738199385273632155144207714, 11.43843241121292595298890662752, 11.86405326257156272124322486984, 12.38098109882178152909384304628, 12.61561510748546313828220353693, 13.15489901751515059955542535017, 13.93865406482178534730604431979

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