Properties

Label 4-162e2-1.1-c1e2-0-7
Degree $4$
Conductor $26244$
Sign $1$
Analytic cond. $1.67334$
Root an. cond. $1.13735$
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 + 3·5-s + 4·7-s − 8-s + 3·10-s + 13-s + 4·14-s − 16-s − 6·17-s − 8·19-s + 5·25-s + 26-s − 9·29-s + 4·31-s − 6·34-s + 12·35-s − 2·37-s − 8·38-s − 3·40-s − 6·41-s − 8·43-s + 12·47-s + 7·49-s + 5·50-s − 12·53-s − 4·56-s − 9·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.34·5-s + 1.51·7-s − 0.353·8-s + 0.948·10-s + 0.277·13-s + 1.06·14-s − 1/4·16-s − 1.45·17-s − 1.83·19-s + 25-s + 0.196·26-s − 1.67·29-s + 0.718·31-s − 1.02·34-s + 2.02·35-s − 0.328·37-s − 1.29·38-s − 0.474·40-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 49-s + 0.707·50-s − 1.64·53-s − 0.534·56-s − 1.18·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(26244\)    =    \(2^{2} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.67334\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{162} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 26244,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.146657453\)
\(L(\frac12)\) \(\approx\) \(2.146657453\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−13.01340994152116054431240305845, −12.93429128948187488972994131825, −12.26625879428020869388346597525, −11.48119090564304142399255937314, −11.22792669909327103848765277347, −10.66866828775986187375433701753, −10.28413299592671584601539887481, −9.550142578397534869870381570351, −8.820286420381976713018688796114, −8.739907206134921971723465010490, −8.021285306323536167733686041875, −7.27722457938436981513682693788, −6.40240599949394411915951835523, −6.23401038195019494050085573043, −5.41192272481053456547808198414, −4.81645177201045047128269588789, −4.45277659599994493652351858628, −3.53881614318847338394124695872, −2.15987477796209772498727713910, −1.91882084049818752724148443255, 1.91882084049818752724148443255, 2.15987477796209772498727713910, 3.53881614318847338394124695872, 4.45277659599994493652351858628, 4.81645177201045047128269588789, 5.41192272481053456547808198414, 6.23401038195019494050085573043, 6.40240599949394411915951835523, 7.27722457938436981513682693788, 8.021285306323536167733686041875, 8.739907206134921971723465010490, 8.820286420381976713018688796114, 9.550142578397534869870381570351, 10.28413299592671584601539887481, 10.66866828775986187375433701753, 11.22792669909327103848765277347, 11.48119090564304142399255937314, 12.26625879428020869388346597525, 12.93429128948187488972994131825, 13.01340994152116054431240305845

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