Properties

Label 4-162e2-1.1-c1e2-0-5
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3·5-s + 7-s − 8-s + 3·10-s − 2·13-s + 14-s − 16-s + 3·17-s − 2·19-s + 5·25-s − 2·26-s − 8·31-s + 3·34-s + 3·35-s + 4·37-s − 2·38-s − 3·40-s + 3·41-s − 2·43-s + 3·47-s − 2·49-s + 5·50-s + 6·53-s − 56-s − 17·61-s − 8·62-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.34·5-s + 0.377·7-s − 0.353·8-s + 0.948·10-s − 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.727·17-s − 0.458·19-s + 25-s − 0.392·26-s − 1.43·31-s + 0.514·34-s + 0.507·35-s + 0.657·37-s − 0.324·38-s − 0.474·40-s + 0.468·41-s − 0.304·43-s + 0.437·47-s − 2/7·49-s + 0.707·50-s + 0.824·53-s − 0.133·56-s − 2.17·61-s − 1.01·62-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: \(0\)
Selberg data: \((4,\ 26244,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.015918253\)
\(L(\frac12)\) \(\approx\) \(2.015918253\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$D_{4}$ \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 60 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
41$D_{4}$ \( 1 - 3 T - 23 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 3 T - p T^{2} - 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$D_{4}$ \( 1 + 14 T + 156 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T - 39 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 5 T + 81 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 12 T + 124 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 13 T + 207 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
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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

−15.0924835796, −14.9008077648, −14.3842501853, −14.1183743309, −13.5380285186, −13.1762231336, −12.7508717346, −12.2206116198, −11.8245133300, −11.0798041997, −10.6297217493, −10.0885893224, −9.65254842649, −9.00502997176, −8.76427689750, −7.71763695356, −7.42743752596, −6.55195943646, −5.92179520590, −5.61837670395, −4.90070955562, −4.35428106074, −3.38224952032, −2.54241019486, −1.64738007120, 1.64738007120, 2.54241019486, 3.38224952032, 4.35428106074, 4.90070955562, 5.61837670395, 5.92179520590, 6.55195943646, 7.42743752596, 7.71763695356, 8.76427689750, 9.00502997176, 9.65254842649, 10.0885893224, 10.6297217493, 11.0798041997, 11.8245133300, 12.2206116198, 12.7508717346, 13.1762231336, 13.5380285186, 14.1183743309, 14.3842501853, 14.9008077648, 15.0924835796

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