Properties

Label 2-3630-1.1-c1-0-20
Degree $2$
Conductor $3630$
Sign $1$
Analytic cond. $28.9856$
Root an. cond. $5.38383$
Motivic weight $1$
Arithmetic yes
Rational no
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 + 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: $1$
Analytic conductor: \(28.9856\)
Root analytic conductor: \(5.38383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3630,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.864187784\)
\(L(\frac12)\) \(\approx\) \(1.864187784\)
\(L(\frac{3}{2})\) 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

−8.290880940125742480169194342953, −7.940862215745849590257436088385, −7.44330304873456599546957259680, −6.60762056841863508804697227334, −5.42571212220391165716797986772, −4.74083250715729700734255420967, −3.84816005652483532754816466871, −2.77367095334957668367721959856, −1.93519020991660845067417571667, −0.896748755987244239299832428147, 0.896748755987244239299832428147, 1.93519020991660845067417571667, 2.77367095334957668367721959856, 3.84816005652483532754816466871, 4.74083250715729700734255420967, 5.42571212220391165716797986772, 6.60762056841863508804697227334, 7.44330304873456599546957259680, 7.940862215745849590257436088385, 8.290880940125742480169194342953

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