Properties

Label 2-6762-1.1-c1-0-91
Degree $2$
Conductor $6762$
Sign $-1$
Analytic cond. $53.9948$
Root an. cond. $7.34811$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 3·5-s − 6-s − 8-s + 9-s + 3·10-s + 4·11-s + 12-s − 3·13-s − 3·15-s + 16-s − 18-s − 3·20-s − 4·22-s − 23-s − 24-s + 4·25-s + 3·26-s + 27-s + 29-s + 3·30-s + 2·31-s − 32-s + 4·33-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 1.20·11-s + 0.288·12-s − 0.832·13-s − 0.774·15-s + 1/4·16-s − 0.235·18-s − 0.670·20-s − 0.852·22-s − 0.208·23-s − 0.204·24-s + 4/5·25-s + 0.588·26-s + 0.192·27-s + 0.185·29-s + 0.547·30-s + 0.359·31-s − 0.176·32-s + 0.696·33-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6762\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(53.9948\)
Root analytic conductor: \(7.34811\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{6762} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6762,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
23 \( 1 + T \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−7.72984510005466926767635591493, −7.00682403300823758782944348276, −6.72930765159729369050500891741, −5.51847013989405402076733526708, −4.51403745497681300670901690347, −3.86658328874784466129529075099, −3.21575754090618482225674584293, −2.23074455193604380094750369526, −1.18108152534614209277844902025, 0, 1.18108152534614209277844902025, 2.23074455193604380094750369526, 3.21575754090618482225674584293, 3.86658328874784466129529075099, 4.51403745497681300670901690347, 5.51847013989405402076733526708, 6.72930765159729369050500891741, 7.00682403300823758782944348276, 7.72984510005466926767635591493

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