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 − 1.58·7-s + 8-s + 9-s − 10-s + 12-s − 5.75·13-s − 1.58·14-s − 15-s + 16-s + 7.17·17-s + 18-s + 1.32·19-s − 20-s − 1.58·21-s − 0.358·23-s + 24-s + 25-s − 5.75·26-s + 27-s − 1.58·28-s + 9.67·29-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 − 0.597·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.288·12-s − 1.59·13-s − 0.422·14-s − 0.258·15-s + 0.250·16-s + 1.74·17-s + 0.235·18-s + 0.303·19-s − 0.223·20-s − 0.344·21-s − 0.0748·23-s + 0.204·24-s + 0.200·25-s − 1.12·26-s + 0.192·27-s − 0.298·28-s + 1.79·29-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

\(L(1)\) \(\approx\) \(3.235206643\)
\(L(\frac12)\) \(\approx\) \(3.235206643\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.58T + 7T^{2} \)
13 \( 1 + 5.75T + 13T^{2} \)
17 \( 1 - 7.17T + 17T^{2} \)
19 \( 1 - 1.32T + 19T^{2} \)
23 \( 1 + 0.358T + 23T^{2} \)
29 \( 1 - 9.67T + 29T^{2} \)
31 \( 1 - 7.19T + 31T^{2} \)
37 \( 1 - 9.73T + 37T^{2} \)
41 \( 1 + 12.0T + 41T^{2} \)
43 \( 1 + 6.41T + 43T^{2} \)
47 \( 1 - 6.52T + 47T^{2} \)
53 \( 1 - 2.34T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 5.63T + 61T^{2} \)
67 \( 1 + 5.07T + 67T^{2} \)
71 \( 1 - 9.11T + 71T^{2} \)
73 \( 1 - 6.94T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 - 7.16T + 89T^{2} \)
97 \( 1 + 1.92T + 97T^{2} \)
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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

−8.173429326086147275438761513690, −7.917281725476302810163513499119, −6.96573168011861146736537091466, −6.46212850784327497591005516285, −5.29231075629187107178354167482, −4.77171184154178775765374398901, −3.78064925181591672341798027975, −3.04251636807936627189316320927, −2.44126164499413824530116810989, −0.932887620285584336683531344424, 0.932887620285584336683531344424, 2.44126164499413824530116810989, 3.04251636807936627189316320927, 3.78064925181591672341798027975, 4.77171184154178775765374398901, 5.29231075629187107178354167482, 6.46212850784327497591005516285, 6.96573168011861146736537091466, 7.917281725476302810163513499119, 8.173429326086147275438761513690

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