Properties

Label 2-259182-1.1-c1-0-119
Degree $2$
Conductor $259182$
Sign $-1$
Analytic cond. $2069.57$
Root an. cond. $45.4926$
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 + 4-s − 2·5-s + 7-s + 8-s − 2·10-s + 2·13-s + 14-s + 16-s − 17-s + 8·19-s − 2·20-s − 8·23-s − 25-s + 2·26-s + 28-s − 2·29-s − 4·31-s + 32-s − 34-s − 2·35-s + 6·37-s + 8·38-s − 2·40-s − 10·41-s + 4·43-s − 8·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 0.632·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 1.83·19-s − 0.447·20-s − 1.66·23-s − 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.371·29-s − 0.718·31-s + 0.176·32-s − 0.171·34-s − 0.338·35-s + 0.986·37-s + 1.29·38-s − 0.316·40-s − 1.56·41-s + 0.609·43-s − 1.17·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(259182\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(2069.57\)
Root analytic conductor: \(45.4926\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 259182,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.27101188749962, −12.40107240193375, −12.05993941435635, −11.74723300832036, −11.44215134983126, −10.80516195941669, −10.51716001938913, −9.778852912774661, −9.428161000070805, −8.790310429930686, −8.159652598453441, −7.905681841271143, −7.281668460722490, −7.146478729406596, −6.276695124106920, −5.757125415745089, −5.510571517963980, −4.804811704454718, −4.250368942927149, −3.837583173703587, −3.473610319648311, −2.817230901372638, −2.149660400386703, −1.532341618639478, −0.8535583170031585, 0, 0.8535583170031585, 1.532341618639478, 2.149660400386703, 2.817230901372638, 3.473610319648311, 3.837583173703587, 4.250368942927149, 4.804811704454718, 5.510571517963980, 5.757125415745089, 6.276695124106920, 7.146478729406596, 7.281668460722490, 7.905681841271143, 8.159652598453441, 8.790310429930686, 9.428161000070805, 9.778852912774661, 10.51716001938913, 10.80516195941669, 11.44215134983126, 11.74723300832036, 12.05993941435635, 12.40107240193375, 13.27101188749962

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