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 + 1.23·7-s + 8-s + 9-s − 10-s − 12-s + 5.61·13-s + 1.23·14-s + 15-s + 16-s + 5.09·17-s + 18-s − 3.23·19-s − 20-s − 1.23·21-s + 0.145·23-s − 24-s + 25-s + 5.61·26-s − 27-s + 1.23·28-s − 0.381·29-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 + 0.467·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.288·12-s + 1.55·13-s + 0.330·14-s + 0.258·15-s + 0.250·16-s + 1.23·17-s + 0.235·18-s − 0.742·19-s − 0.223·20-s − 0.269·21-s + 0.0304·23-s − 0.204·24-s + 0.200·25-s + 1.10·26-s − 0.192·27-s + 0.233·28-s − 0.0709·29-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

\(L(1)\) \(\approx\) \(2.666900801\)
\(L(\frac12)\) \(\approx\) \(2.666900801\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.23T + 7T^{2} \)
13 \( 1 - 5.61T + 13T^{2} \)
17 \( 1 - 5.09T + 17T^{2} \)
19 \( 1 + 3.23T + 19T^{2} \)
23 \( 1 - 0.145T + 23T^{2} \)
29 \( 1 + 0.381T + 29T^{2} \)
31 \( 1 + 3.61T + 31T^{2} \)
37 \( 1 + 8.61T + 37T^{2} \)
41 \( 1 + 3.23T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 + 0.472T + 53T^{2} \)
59 \( 1 - 2.90T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 6.09T + 67T^{2} \)
71 \( 1 - 1.52T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 - 7.09T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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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

−8.432558793556105026022099598104, −7.67493544865306120158796026037, −6.98464193618107567478892230879, −6.07473107019606234201159244709, −5.60580644063062789793460646558, −4.74833463073566246059108095322, −3.90004228016573461707863764169, −3.35193610237345354544003971592, −1.96234430021605036967134723404, −0.932696976675516260579889163180, 0.932696976675516260579889163180, 1.96234430021605036967134723404, 3.35193610237345354544003971592, 3.90004228016573461707863764169, 4.74833463073566246059108095322, 5.60580644063062789793460646558, 6.07473107019606234201159244709, 6.98464193618107567478892230879, 7.67493544865306120158796026037, 8.432558793556105026022099598104

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