Properties

Label 2-6762-1.1-c1-0-90
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 1.82·5-s + 6-s + 8-s + 9-s + 1.82·10-s + 1.68·11-s + 12-s + 3.77·13-s + 1.82·15-s + 16-s + 2.59·17-s + 18-s + 1.09·19-s + 1.82·20-s + 1.68·22-s + 23-s + 24-s − 1.65·25-s + 3.77·26-s + 27-s − 0.771·29-s + 1.82·30-s − 3.86·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.817·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.578·10-s + 0.506·11-s + 0.288·12-s + 1.04·13-s + 0.472·15-s + 0.250·16-s + 0.630·17-s + 0.235·18-s + 0.250·19-s + 0.408·20-s + 0.358·22-s + 0.208·23-s + 0.204·24-s − 0.331·25-s + 0.739·26-s + 0.192·27-s − 0.143·29-s + 0.333·30-s − 0.693·31-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: $\chi_{6762} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6762,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.589916325\)
\(L(\frac12)\) \(\approx\) \(5.589916325\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.82T + 5T^{2} \)
11 \( 1 - 1.68T + 11T^{2} \)
13 \( 1 - 3.77T + 13T^{2} \)
17 \( 1 - 2.59T + 17T^{2} \)
19 \( 1 - 1.09T + 19T^{2} \)
29 \( 1 + 0.771T + 29T^{2} \)
31 \( 1 + 3.86T + 31T^{2} \)
37 \( 1 - 2.45T + 37T^{2} \)
41 \( 1 + 1.09T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + 2.22T + 47T^{2} \)
53 \( 1 + 7.37T + 53T^{2} \)
59 \( 1 + 6.42T + 59T^{2} \)
61 \( 1 - 9.08T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 - 8.22T + 71T^{2} \)
73 \( 1 + 2.86T + 73T^{2} \)
79 \( 1 - 0.771T + 79T^{2} \)
83 \( 1 - 3.33T + 83T^{2} \)
89 \( 1 - 3.20T + 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−7.88168652001422038729722045092, −7.24789275830200066871279991787, −6.36116212031059271862711970592, −5.90603460714836171437570992063, −5.20463603078767512602805686309, −4.23457163350382627171843557473, −3.59988632847214976172859410917, −2.85880406239217141672989474859, −1.90830249315047160082654340972, −1.17850285262045586183958881755, 1.17850285262045586183958881755, 1.90830249315047160082654340972, 2.85880406239217141672989474859, 3.59988632847214976172859410917, 4.23457163350382627171843557473, 5.20463603078767512602805686309, 5.90603460714836171437570992063, 6.36116212031059271862711970592, 7.24789275830200066871279991787, 7.88168652001422038729722045092

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