Properties

Label 2-327990-1.1-c1-0-37
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 + 2·7-s + 8-s + 9-s − 10-s − 12-s + 13-s + 2·14-s + 15-s + 16-s + 6·17-s + 18-s − 2·19-s − 20-s − 2·21-s + 3·23-s − 24-s + 25-s + 26-s − 27-s + 2·28-s + 30-s − 5·31-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.277·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.436·21-s + 0.625·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.377·28-s + 0.182·30-s − 0.898·31-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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−12.72036972228836, −12.36457155110405, −12.00975248461726, −11.44682667609800, −11.16348209992041, −10.59676378433539, −10.48453433210755, −9.654659130915967, −9.237015518445213, −8.625888188691378, −8.066912032884559, −7.653442393536859, −7.309535127343400, −6.731031472706050, −6.204470864631519, −5.645630861213592, −5.325788268739190, −4.843607678852983, −4.302675408572086, −3.818100178078858, −3.364728409825004, −2.707932915150288, −2.060391956043430, −1.330967584419034, −0.9595653842424412, 0, 0.9595653842424412, 1.330967584419034, 2.060391956043430, 2.707932915150288, 3.364728409825004, 3.818100178078858, 4.302675408572086, 4.843607678852983, 5.325788268739190, 5.645630861213592, 6.204470864631519, 6.731031472706050, 7.309535127343400, 7.653442393536859, 8.066912032884559, 8.625888188691378, 9.237015518445213, 9.654659130915967, 10.48453433210755, 10.59676378433539, 11.16348209992041, 11.44682667609800, 12.00975248461726, 12.36457155110405, 12.72036972228836

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