Properties

Label 2-327990-1.1-c1-0-34
Degree $2$
Conductor $327990$
Sign $-1$
Analytic cond. $2619.01$
Root an. cond. $51.1762$
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-s + 4-s + 5-s − 6-s + 5·7-s + 8-s + 9-s + 10-s − 11-s − 12-s + 13-s + 5·14-s − 15-s + 16-s − 2·17-s + 18-s − 4·19-s + 20-s − 5·21-s − 22-s − 9·23-s − 24-s + 25-s + 26-s − 27-s + 5·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.88·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.277·13-s + 1.33·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s − 1.09·21-s − 0.213·22-s − 1.87·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.944·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(327990\)    =    \(2 \cdot 3 \cdot 5 \cdot 13 \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(2619.01\)
Root analytic conductor: \(51.1762\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{327990} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 327990,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 - T \)
29 \( 1 \)
good7 \( 1 - 5 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−12.73424322649077, −12.27126681534909, −11.91118373582323, −11.59710667328697, −11.07494710537239, −10.55902260709437, −10.38473293499595, −9.933681397666289, −9.023266491274610, −8.613404002746677, −8.148116545856873, −7.819365541570082, −7.191949558855708, −6.606230591710034, −6.275109301027204, −5.579802546597625, −5.376056277628222, −4.764880162500319, −4.405971337342258, −3.996812654468363, −3.304694297779064, −2.428150087412158, −1.894628413748443, −1.760952855758371, −0.9155121465608693, 0, 0.9155121465608693, 1.760952855758371, 1.894628413748443, 2.428150087412158, 3.304694297779064, 3.996812654468363, 4.405971337342258, 4.764880162500319, 5.376056277628222, 5.579802546597625, 6.275109301027204, 6.606230591710034, 7.191949558855708, 7.819365541570082, 8.148116545856873, 8.613404002746677, 9.023266491274610, 9.933681397666289, 10.38473293499595, 10.55902260709437, 11.07494710537239, 11.59710667328697, 11.91118373582323, 12.27126681534909, 12.73424322649077

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