Properties

Degree $2$
Conductor $3630$
Sign $1$
Motivic weight $1$
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 − 5.10·7-s + 8-s + 9-s − 10-s + 12-s − 1.33·13-s − 5.10·14-s − 15-s + 16-s − 0.775·17-s + 18-s + 0.0785·19-s − 20-s − 5.10·21-s + 6.64·23-s + 24-s + 25-s − 1.33·26-s + 27-s − 5.10·28-s + 2.01·29-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 − 1.93·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.288·12-s − 0.369·13-s − 1.36·14-s − 0.258·15-s + 0.250·16-s − 0.188·17-s + 0.235·18-s + 0.0180·19-s − 0.223·20-s − 1.11·21-s + 1.38·23-s + 0.204·24-s + 0.200·25-s − 0.261·26-s + 0.192·27-s − 0.965·28-s + 0.374·29-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: \(0\)
Selberg data: \((2,\ 3630,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.640398045\)
\(L(\frac12)\) \(\approx\) \(2.640398045\)
\(L(\frac{3}{2})\) 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 \( 1 + 5.10T + 7T^{2} \)
13 \( 1 + 1.33T + 13T^{2} \)
17 \( 1 + 0.775T + 17T^{2} \)
19 \( 1 - 0.0785T + 19T^{2} \)
23 \( 1 - 6.64T + 23T^{2} \)
29 \( 1 - 2.01T + 29T^{2} \)
31 \( 1 - 8.49T + 31T^{2} \)
37 \( 1 + 3.93T + 37T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 - 6.01T + 43T^{2} \)
47 \( 1 - 6.56T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + 5.32T + 59T^{2} \)
61 \( 1 - 3.74T + 61T^{2} \)
67 \( 1 - 0.588T + 67T^{2} \)
71 \( 1 + 2.31T + 71T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + 9.67T + 89T^{2} \)
97 \( 1 + 13.4T + 97T^{2} \)
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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

−8.684559744654259073913034109185, −7.50984418912049816810747819580, −7.05395160617087736945323465837, −6.35969901416536659998850561885, −5.60933914048011188207245608817, −4.48077183998019012245543089234, −3.86562410746253465151529381445, −2.90196480188723086541213408767, −2.64646573201126995977151711303, −0.815355966158160828379332981311, 0.815355966158160828379332981311, 2.64646573201126995977151711303, 2.90196480188723086541213408767, 3.86562410746253465151529381445, 4.48077183998019012245543089234, 5.60933914048011188207245608817, 6.35969901416536659998850561885, 7.05395160617087736945323465837, 7.50984418912049816810747819580, 8.684559744654259073913034109185

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