Properties

Label 2-6762-1.1-c1-0-114
Degree $2$
Conductor $6762$
Sign $-1$
Analytic cond. $53.9948$
Root an. cond. $7.34811$
Motivic weight $1$
Arithmetic yes
Rational no
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 + 2.41·5-s + 6-s − 8-s + 9-s − 2.41·10-s + 2·11-s − 12-s − 3.82·13-s − 2.41·15-s + 16-s − 5.24·17-s − 18-s + 7.65·19-s + 2.41·20-s − 2·22-s + 23-s + 24-s + 0.828·25-s + 3.82·26-s − 27-s − 6.82·29-s + 2.41·30-s + 2·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.07·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.763·10-s + 0.603·11-s − 0.288·12-s − 1.06·13-s − 0.623·15-s + 0.250·16-s − 1.27·17-s − 0.235·18-s + 1.75·19-s + 0.539·20-s − 0.426·22-s + 0.208·23-s + 0.204·24-s + 0.165·25-s + 0.750·26-s − 0.192·27-s − 1.26·29-s + 0.440·30-s + 0.359·31-s − 0.176·32-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: no
Arithmetic: yes
Character: Trivial
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 \( 1 - 2.41T + 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 3.82T + 13T^{2} \)
17 \( 1 + 5.24T + 17T^{2} \)
19 \( 1 - 7.65T + 19T^{2} \)
29 \( 1 + 6.82T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 8.82T + 37T^{2} \)
41 \( 1 - 5.17T + 41T^{2} \)
43 \( 1 + 0.343T + 43T^{2} \)
47 \( 1 + 9T + 47T^{2} \)
53 \( 1 - 4.07T + 53T^{2} \)
59 \( 1 - 9.65T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 4.89T + 67T^{2} \)
71 \( 1 + 4.17T + 71T^{2} \)
73 \( 1 + 0.656T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 7.17T + 83T^{2} \)
89 \( 1 + 6.82T + 89T^{2} \)
97 \( 1 + 1.65T + 97T^{2} \)
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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

−7.35555364930301269473262971884, −7.06637445993770824016970542142, −6.29114639329103663611588379297, −5.54433240816924753327134644685, −5.06191074620854839737914845218, −4.02420443094422676565869295832, −2.90946007783184454038611896724, −2.03371527477243769959777579327, −1.30153165967718997868510119324, 0, 1.30153165967718997868510119324, 2.03371527477243769959777579327, 2.90946007783184454038611896724, 4.02420443094422676565869295832, 5.06191074620854839737914845218, 5.54433240816924753327134644685, 6.29114639329103663611588379297, 7.06637445993770824016970542142, 7.35555364930301269473262971884

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