Properties

Label 2-259182-1.1-c1-0-142
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 + 7·13-s + 14-s + 16-s + 17-s + 6·19-s − 2·20-s + 3·23-s − 25-s + 7·26-s + 28-s − 9·29-s − 11·31-s + 32-s + 34-s − 2·35-s + 4·37-s + 6·38-s − 2·40-s + 3·41-s − 43-s + 3·46-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 + 1.94·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 1.37·19-s − 0.447·20-s + 0.625·23-s − 1/5·25-s + 1.37·26-s + 0.188·28-s − 1.67·29-s − 1.97·31-s + 0.176·32-s + 0.171·34-s − 0.338·35-s + 0.657·37-s + 0.973·38-s − 0.316·40-s + 0.468·41-s − 0.152·43-s + 0.442·46-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: $\chi_{259182} (1, \cdot )$
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 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 11 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.17658596496996, −12.49159251303062, −12.23859602845277, −11.54453140907341, −11.22358172219180, −10.97389713203628, −10.67482196945859, −9.662354369596051, −9.376735100907067, −8.846881807431495, −8.243022997201756, −7.778366229482957, −7.447220536045043, −7.021069663133198, −6.244695051178899, −5.882759857698171, −5.335815070531113, −4.994405271474781, −4.099168952709129, −3.866281814089171, −3.402660823588199, −3.010684371469238, −2.032206099333891, −1.478887947984782, −0.9597794771470126, 0, 0.9597794771470126, 1.478887947984782, 2.032206099333891, 3.010684371469238, 3.402660823588199, 3.866281814089171, 4.099168952709129, 4.994405271474781, 5.335815070531113, 5.882759857698171, 6.244695051178899, 7.021069663133198, 7.447220536045043, 7.778366229482957, 8.243022997201756, 8.846881807431495, 9.376735100907067, 9.662354369596051, 10.67482196945859, 10.97389713203628, 11.22358172219180, 11.54453140907341, 12.23859602845277, 12.49159251303062, 13.17658596496996

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