Properties

Degree $2$
Conductor $3630$
Sign $-1$
Motivic weight $1$
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 − 0.267·7-s + 8-s + 9-s − 10-s + 12-s − 4.46·13-s − 0.267·14-s − 15-s + 16-s − 1.73·17-s + 18-s − 7·19-s − 20-s − 0.267·21-s − 6.73·23-s + 24-s + 25-s − 4.46·26-s + 27-s − 0.267·28-s − 5·29-s − 30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 0.101·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.288·12-s − 1.23·13-s − 0.0716·14-s − 0.258·15-s + 0.250·16-s − 0.420·17-s + 0.235·18-s − 1.60·19-s − 0.223·20-s − 0.0584·21-s − 1.40·23-s + 0.204·24-s + 0.200·25-s − 0.875·26-s + 0.192·27-s − 0.0506·28-s − 0.928·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3630\)    =    \(2 \cdot 3 \cdot 5 \cdot 11^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{3630} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3630,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
11 \( 1 \)
good7 \( 1 + 0.267T + 7T^{2} \)
13 \( 1 + 4.46T + 13T^{2} \)
17 \( 1 + 1.73T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + 6.73T + 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 - 6.19T + 31T^{2} \)
37 \( 1 + 7.19T + 37T^{2} \)
41 \( 1 - 4.19T + 41T^{2} \)
43 \( 1 - 2.19T + 43T^{2} \)
47 \( 1 - 4.92T + 47T^{2} \)
53 \( 1 - 0.732T + 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 + 15.1T + 61T^{2} \)
67 \( 1 - 4.53T + 67T^{2} \)
71 \( 1 + 2.46T + 71T^{2} \)
73 \( 1 - 9.66T + 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + 9.53T + 83T^{2} \)
89 \( 1 - 9.46T + 89T^{2} \)
97 \( 1 - 8.73T + 97T^{2} \)
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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

−7.987380278764521617704016378298, −7.51653426498253516174255243545, −6.62767653326035805056188088007, −6.01112544186951641746073845838, −4.85414793444078759597647877069, −4.32802134742402898192968693657, −3.56200413906123759278691398321, −2.55476549388887777358395752590, −1.89630619936456074985842817717, 0, 1.89630619936456074985842817717, 2.55476549388887777358395752590, 3.56200413906123759278691398321, 4.32802134742402898192968693657, 4.85414793444078759597647877069, 6.01112544186951641746073845838, 6.62767653326035805056188088007, 7.51653426498253516174255243545, 7.987380278764521617704016378298

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