Properties

Label 2-6762-1.1-c1-0-9
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 12-s − 2·13-s + 16-s − 18-s − 2·19-s − 23-s + 24-s − 5·25-s + 2·26-s − 27-s − 6·29-s + 4·31-s − 32-s + 36-s − 10·37-s + 2·38-s + 2·39-s + 6·41-s + 2·43-s + 46-s − 48-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.235·18-s − 0.458·19-s − 0.208·23-s + 0.204·24-s − 25-s + 0.392·26-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 1/6·36-s − 1.64·37-s + 0.324·38-s + 0.320·39-s + 0.937·41-s + 0.304·43-s + 0.147·46-s − 0.144·48-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6762,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7368266725\)
\(L(\frac12)\) \(\approx\) \(0.7368266725\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.931404083423785641892288585422, −7.29332377024181140322458552093, −6.71487145605258108426933912760, −5.87847729116976347092194654844, −5.35805936549294925945573045992, −4.37505316682269263615457205700, −3.61762330685091804253450437465, −2.46892708556937628232716387473, −1.71676922593457090451251489728, −0.49716895631365862715746178734, 0.49716895631365862715746178734, 1.71676922593457090451251489728, 2.46892708556937628232716387473, 3.61762330685091804253450437465, 4.37505316682269263615457205700, 5.35805936549294925945573045992, 5.87847729116976347092194654844, 6.71487145605258108426933912760, 7.29332377024181140322458552093, 7.931404083423785641892288585422

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