Properties

Label 4-162e2-1.1-c1e2-0-8
Degree $4$
Conductor $26244$
Sign $-1$
Analytic cond. $1.67334$
Root an. cond. $1.13735$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 2·7-s − 5·13-s + 4·16-s − 2·19-s − 25-s + 4·28-s − 8·31-s − 5·37-s − 20·43-s + 7·49-s + 10·52-s − 5·61-s − 8·64-s − 2·67-s + 73-s + 4·76-s + 10·79-s + 10·91-s + 10·97-s + 2·100-s + 101-s + 103-s + 107-s + 109-s − 8·112-s + 113-s + ⋯
L(s)  = 1  − 4-s − 0.755·7-s − 1.38·13-s + 16-s − 0.458·19-s − 1/5·25-s + 0.755·28-s − 1.43·31-s − 0.821·37-s − 3.04·43-s + 49-s + 1.38·52-s − 0.640·61-s − 64-s − 0.244·67-s + 0.117·73-s + 0.458·76-s + 1.12·79-s + 1.04·91-s + 1.01·97-s + 1/5·100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s − 0.755·112-s + 0.0940·113-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: $\chi_{26244} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 26244,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T^{2} \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2^2$ \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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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

−15.5661348379, −14.9978425101, −14.8618552521, −14.2503916444, −13.7566908660, −13.3063109520, −12.8913822048, −12.4454649925, −12.0211228594, −11.5204763615, −10.6345411192, −10.2908491834, −9.78188532983, −9.37050253492, −8.85577608372, −8.32250088323, −7.66007083372, −7.10233129189, −6.52504595094, −5.73730543418, −5.11207898163, −4.64259559899, −3.72482723315, −3.18716773167, −1.98532177689, 0, 1.98532177689, 3.18716773167, 3.72482723315, 4.64259559899, 5.11207898163, 5.73730543418, 6.52504595094, 7.10233129189, 7.66007083372, 8.32250088323, 8.85577608372, 9.37050253492, 9.78188532983, 10.2908491834, 10.6345411192, 11.5204763615, 12.0211228594, 12.4454649925, 12.8913822048, 13.3063109520, 13.7566908660, 14.2503916444, 14.8618552521, 14.9978425101, 15.5661348379

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