Properties

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

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 − 4·11-s + 12-s − 13-s − 2·14-s + 15-s + 16-s − 4·17-s + 18-s + 2·19-s + 20-s − 2·21-s − 4·22-s + 2·23-s + 24-s + 25-s − 26-s + 27-s − 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.458·19-s + 0.223·20-s − 0.436·21-s − 0.852·22-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 + ⋯

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: \(0\)
Selberg data: \((2,\ 327990,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.640681941\)
\(L(\frac12)\) \(\approx\) \(1.640681941\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−12.74887602038901, −12.39413361827576, −11.73394097821428, −11.33566408765655, −10.63363240078799, −10.38465195893515, −9.966268270784220, −9.432998453024011, −8.899994518145176, −8.580467828768712, −7.890749023057921, −7.426331461583663, −7.011935637774488, −6.517106332858639, −6.099102912658800, −5.400784637021377, −5.018285750465511, −4.696024148304800, −3.845119194662653, −3.436044851372652, −2.908282487291769, −2.528319205714543, −1.905557011993656, −1.398684220619743, −0.2567666859720706, 0.2567666859720706, 1.398684220619743, 1.905557011993656, 2.528319205714543, 2.908282487291769, 3.436044851372652, 3.845119194662653, 4.696024148304800, 5.018285750465511, 5.400784637021377, 6.099102912658800, 6.517106332858639, 7.011935637774488, 7.426331461583663, 7.890749023057921, 8.580467828768712, 8.899994518145176, 9.432998453024011, 9.966268270784220, 10.38465195893515, 10.63363240078799, 11.33566408765655, 11.73394097821428, 12.39413361827576, 12.74887602038901

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