Properties

Label 2-6762-1.1-c1-0-62
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 + 11-s + 12-s − 6·13-s + 16-s + 5·17-s + 18-s + 6·19-s + 22-s + 23-s + 24-s − 5·25-s − 6·26-s + 27-s + 9·29-s − 6·31-s + 32-s + 33-s + 5·34-s + 36-s + 6·37-s + 6·38-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 + 0.301·11-s + 0.288·12-s − 1.66·13-s + 1/4·16-s + 1.21·17-s + 0.235·18-s + 1.37·19-s + 0.213·22-s + 0.208·23-s + 0.204·24-s − 25-s − 1.17·26-s + 0.192·27-s + 1.67·29-s − 1.07·31-s + 0.176·32-s + 0.174·33-s + 0.857·34-s + 1/6·36-s + 0.986·37-s + 0.973·38-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

\(L(1)\) \(\approx\) \(4.270163485\)
\(L(\frac12)\) \(\approx\) \(4.270163485\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−7.65694925147766315429888627593, −7.41695611686945517688052630184, −6.66052328308385928849932942971, −5.62235681323745468638743858240, −5.17059702877625050156504531728, −4.34831853743474764106439445263, −3.52405513181818823316538020988, −2.86394011695279586144184033322, −2.09338309381785487030637769641, −0.948360650385425359388273894828, 0.948360650385425359388273894828, 2.09338309381785487030637769641, 2.86394011695279586144184033322, 3.52405513181818823316538020988, 4.34831853743474764106439445263, 5.17059702877625050156504531728, 5.62235681323745468638743858240, 6.66052328308385928849932942971, 7.41695611686945517688052630184, 7.65694925147766315429888627593

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