Properties

Label 2-338130-1.1-c1-0-34
Degree $2$
Conductor $338130$
Sign $-1$
Analytic cond. $2699.98$
Root an. cond. $51.9613$
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 − 5-s − 2·7-s − 8-s + 10-s + 13-s + 2·14-s + 16-s − 19-s − 20-s − 4·23-s + 25-s − 26-s − 2·28-s − 2·29-s − 8·31-s − 32-s + 2·35-s − 3·37-s + 38-s + 40-s + 5·41-s − 2·43-s + 4·46-s − 9·47-s − 3·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s + 0.277·13-s + 0.534·14-s + 1/4·16-s − 0.229·19-s − 0.223·20-s − 0.834·23-s + 1/5·25-s − 0.196·26-s − 0.377·28-s − 0.371·29-s − 1.43·31-s − 0.176·32-s + 0.338·35-s − 0.493·37-s + 0.162·38-s + 0.158·40-s + 0.780·41-s − 0.304·43-s + 0.589·46-s − 1.31·47-s − 3/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338130\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(2699.98\)
Root analytic conductor: \(51.9613\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{338130} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 338130,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
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

−12.66177064783953, −12.35620911889529, −11.88005896228383, −11.30644494430215, −10.97164806260248, −10.57204345913276, −9.981253935623248, −9.587852915394901, −9.216315393446266, −8.716776547317306, −8.215780647436617, −7.764593386633522, −7.392549628700463, −6.770789364343255, −6.403476525667279, −5.959336381279828, −5.369240133488420, −4.806543918719502, −4.113402928392581, −3.585375190864613, −3.265818865698792, −2.580688102575380, −1.891898284703042, −1.463628632067797, −0.5189194210486801, 0, 0.5189194210486801, 1.463628632067797, 1.891898284703042, 2.580688102575380, 3.265818865698792, 3.585375190864613, 4.113402928392581, 4.806543918719502, 5.369240133488420, 5.959336381279828, 6.403476525667279, 6.770789364343255, 7.392549628700463, 7.764593386633522, 8.215780647436617, 8.716776547317306, 9.216315393446266, 9.587852915394901, 9.981253935623248, 10.57204345913276, 10.97164806260248, 11.30644494430215, 11.88005896228383, 12.35620911889529, 12.66177064783953

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