Properties

Label 2-338130-1.1-c1-0-46
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 + 2·19-s − 20-s − 6·23-s + 25-s − 26-s − 2·28-s + 4·31-s − 32-s + 2·35-s − 2·37-s − 2·38-s + 40-s − 6·41-s − 4·43-s + 6·46-s − 3·49-s − 50-s + 52-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 + 0.458·19-s − 0.223·20-s − 1.25·23-s + 1/5·25-s − 0.196·26-s − 0.377·28-s + 0.718·31-s − 0.176·32-s + 0.338·35-s − 0.328·37-s − 0.324·38-s + 0.158·40-s − 0.937·41-s − 0.609·43-s + 0.884·46-s − 3/7·49-s − 0.141·50-s + 0.138·52-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 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−12.83707876329784, −12.14627589632536, −11.86516248094048, −11.47466817395675, −11.02801993199792, −10.24495762417534, −10.11439112877140, −9.771614759244408, −9.098553329079598, −8.656426743039764, −8.289685832770845, −7.803543891284647, −7.312594703881259, −6.815868068293258, −6.385979886283598, −5.979192230082390, −5.326933793366693, −4.848601406704261, −4.074446752932503, −3.655655751245409, −3.199152406696930, −2.579768932082698, −1.992268433332078, −1.324098283927979, −0.6188171575412270, 0, 0.6188171575412270, 1.324098283927979, 1.992268433332078, 2.579768932082698, 3.199152406696930, 3.655655751245409, 4.074446752932503, 4.848601406704261, 5.326933793366693, 5.979192230082390, 6.385979886283598, 6.815868068293258, 7.312594703881259, 7.803543891284647, 8.289685832770845, 8.656426743039764, 9.098553329079598, 9.771614759244408, 10.11439112877140, 10.24495762417534, 11.02801993199792, 11.47466817395675, 11.86516248094048, 12.14627589632536, 12.83707876329784

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