Properties

Label 4-162e2-1.1-c1e2-0-6
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-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: \(1\)
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$C_2$ \( 1 + T + T^{2} \)
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

−15.7862902551, −15.1125160474, −14.9468167824, −14.4349715262, −13.8650604200, −13.1878356688, −12.8566007330, −12.1869509620, −11.8712513488, −11.2542086161, −10.8792104579, −10.5177693695, −9.72707770243, −9.20999190464, −8.77667401384, −8.15211608460, −7.77630352879, −7.30457154996, −6.71780977851, −5.95930811158, −4.97029775324, −4.53522322811, −3.85481148818, −2.98754987955, −1.77105843432, 0, 1.77105843432, 2.98754987955, 3.85481148818, 4.53522322811, 4.97029775324, 5.95930811158, 6.71780977851, 7.30457154996, 7.77630352879, 8.15211608460, 8.77667401384, 9.20999190464, 9.72707770243, 10.5177693695, 10.8792104579, 11.2542086161, 11.8712513488, 12.1869509620, 12.8566007330, 13.1878356688, 13.8650604200, 14.4349715262, 14.9468167824, 15.1125160474, 15.7862902551

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