Properties

Label 2-327990-1.1-c1-0-43
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 $2$

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 − 4·19-s − 20-s + 2·21-s − 2·23-s − 24-s + 25-s + 26-s − 27-s − 2·28-s + 30-s − 8·31-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 − 0.917·19-s − 0.223·20-s + 0.436·21-s − 0.417·23-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 − 1.43·31-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: \(2\)
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 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−12.98272218073684, −12.70191707772891, −12.15198825998944, −11.81453803193303, −11.15605331712164, −10.97870517636627, −10.51499294600196, −10.03566362858316, −9.400344780541114, −8.909583912741083, −8.572930254794526, −7.821817260867848, −7.370799091718465, −6.907463658881356, −6.428004102172755, −6.063801515282187, −5.694472335242342, −4.878271917950337, −4.564384773093103, −4.067973145618690, −3.567669043338597, −3.090022331802599, −2.304473698473995, −1.915322509208955, −1.113536007169391, 0, 0, 1.113536007169391, 1.915322509208955, 2.304473698473995, 3.090022331802599, 3.567669043338597, 4.067973145618690, 4.564384773093103, 4.878271917950337, 5.694472335242342, 6.063801515282187, 6.428004102172755, 6.907463658881356, 7.370799091718465, 7.821817260867848, 8.572930254794526, 8.909583912741083, 9.400344780541114, 10.03566362858316, 10.51499294600196, 10.97870517636627, 11.15605331712164, 11.81453803193303, 12.15198825998944, 12.70191707772891, 12.98272218073684

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