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 − 1.38·7-s + 8-s + 9-s − 10-s − 12-s + 1.61·13-s − 1.38·14-s + 15-s + 16-s − 2.47·17-s + 18-s − 1.61·19-s − 20-s + 1.38·21-s − 1.85·23-s − 24-s + 25-s + 1.61·26-s − 27-s − 1.38·28-s + 9.70·29-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 − 0.522·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.288·12-s + 0.448·13-s − 0.369·14-s + 0.258·15-s + 0.250·16-s − 0.599·17-s + 0.235·18-s − 0.371·19-s − 0.223·20-s + 0.301·21-s − 0.386·23-s − 0.204·24-s + 0.200·25-s + 0.317·26-s − 0.192·27-s − 0.261·28-s + 1.80·29-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 \( 1 + 1.38T + 7T^{2} \)
13 \( 1 - 1.61T + 13T^{2} \)
17 \( 1 + 2.47T + 17T^{2} \)
19 \( 1 + 1.61T + 19T^{2} \)
23 \( 1 + 1.85T + 23T^{2} \)
29 \( 1 - 9.70T + 29T^{2} \)
31 \( 1 + 4.76T + 31T^{2} \)
37 \( 1 - 1.61T + 37T^{2} \)
41 \( 1 + 6.32T + 41T^{2} \)
43 \( 1 + 5.23T + 43T^{2} \)
47 \( 1 + 0.381T + 47T^{2} \)
53 \( 1 - 5.09T + 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 + 6.76T + 61T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 - 6.94T + 73T^{2} \)
79 \( 1 + 4.47T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + 2.94T + 97T^{2} \)
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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

−8.131396103738664002563947691408, −7.15430403376256866444549091850, −6.54602188916894816412668355635, −6.01230982118239416469352634948, −5.04907235577840709778785839998, −4.38449580434776504274105986292, −3.59920611396912255574573407326, −2.72726370297914536668279575071, −1.48036518583715417267933486074, 0, 1.48036518583715417267933486074, 2.72726370297914536668279575071, 3.59920611396912255574573407326, 4.38449580434776504274105986292, 5.04907235577840709778785839998, 6.01230982118239416469352634948, 6.54602188916894816412668355635, 7.15430403376256866444549091850, 8.131396103738664002563947691408

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