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 − 3.73·7-s + 8-s + 9-s − 10-s + 12-s + 2.46·13-s − 3.73·14-s − 15-s + 16-s + 1.73·17-s + 18-s − 7·19-s − 20-s − 3.73·21-s − 3.26·23-s + 24-s + 25-s + 2.46·26-s + 27-s − 3.73·28-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 − 1.41·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.288·12-s + 0.683·13-s − 0.997·14-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 − 0.814·21-s − 0.681·23-s + 0.204·24-s + 0.200·25-s + 0.483·26-s + 0.192·27-s − 0.705·28-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 \( 1 + 3.73T + 7T^{2} \)
13 \( 1 - 2.46T + 13T^{2} \)
17 \( 1 - 1.73T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + 3.26T + 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + 4.19T + 31T^{2} \)
37 \( 1 - 3.19T + 37T^{2} \)
41 \( 1 + 6.19T + 41T^{2} \)
43 \( 1 + 8.19T + 43T^{2} \)
47 \( 1 + 8.92T + 47T^{2} \)
53 \( 1 + 2.73T + 53T^{2} \)
59 \( 1 - 5.66T + 59T^{2} \)
61 \( 1 - 9.12T + 61T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 - 4.46T + 71T^{2} \)
73 \( 1 + 7.66T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 16.4T + 83T^{2} \)
89 \( 1 - 2.53T + 89T^{2} \)
97 \( 1 - 5.26T + 97T^{2} \)
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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

−8.248390750755783289191677225462, −7.28752323474667045110842603472, −6.58460240435678906851513840738, −6.08310664686536769519996331058, −5.08712811974075621937109589836, −3.90179327274581027441527375593, −3.68594827446393547320995992960, −2.77793729437577847741261455881, −1.74979085625501629176136000626, 0, 1.74979085625501629176136000626, 2.77793729437577847741261455881, 3.68594827446393547320995992960, 3.90179327274581027441527375593, 5.08712811974075621937109589836, 6.08310664686536769519996331058, 6.58460240435678906851513840738, 7.28752323474667045110842603472, 8.248390750755783289191677225462

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