Properties

Label 2-6762-1.1-c1-0-47
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 − 3.82·5-s + 6-s + 8-s + 9-s − 3.82·10-s + 5.27·11-s + 12-s + 4.35·13-s − 3.82·15-s + 16-s − 2.46·17-s + 18-s − 1.92·19-s − 3.82·20-s + 5.27·22-s + 23-s + 24-s + 9.65·25-s + 4.35·26-s + 27-s − 1.35·29-s − 3.82·30-s − 1.43·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.71·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.21·10-s + 1.59·11-s + 0.288·12-s + 1.20·13-s − 0.988·15-s + 0.250·16-s − 0.599·17-s + 0.235·18-s − 0.440·19-s − 0.856·20-s + 1.12·22-s + 0.208·23-s + 0.204·24-s + 1.93·25-s + 0.854·26-s + 0.192·27-s − 0.252·29-s − 0.698·30-s − 0.258·31-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

\(L(1)\) \(\approx\) \(3.359911244\)
\(L(\frac12)\) \(\approx\) \(3.359911244\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.82T + 5T^{2} \)
11 \( 1 - 5.27T + 11T^{2} \)
13 \( 1 - 4.35T + 13T^{2} \)
17 \( 1 + 2.46T + 17T^{2} \)
19 \( 1 + 1.92T + 19T^{2} \)
29 \( 1 + 1.35T + 29T^{2} \)
31 \( 1 + 1.43T + 31T^{2} \)
37 \( 1 - 6.63T + 37T^{2} \)
41 \( 1 - 1.92T + 41T^{2} \)
43 \( 1 + 8.21T + 43T^{2} \)
47 \( 1 + 6.99T + 47T^{2} \)
53 \( 1 + 2.88T + 53T^{2} \)
59 \( 1 - 4.29T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 - 5.33T + 67T^{2} \)
71 \( 1 - 7.64T + 71T^{2} \)
73 \( 1 + 0.437T + 73T^{2} \)
79 \( 1 - 1.35T + 79T^{2} \)
83 \( 1 + 4.37T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 - 0.298T + 97T^{2} \)
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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

−8.120799071850939376041909440426, −7.08895065610519308635593052039, −6.74400648520150631189407193668, −5.98543826379842968287378331746, −4.76214164818281453252373969762, −4.15816002535406300806085538064, −3.69545022082359932926710042491, −3.18151716582751451571947937862, −1.90905534449666536080470170231, −0.850053492920273040032462580801, 0.850053492920273040032462580801, 1.90905534449666536080470170231, 3.18151716582751451571947937862, 3.69545022082359932926710042491, 4.15816002535406300806085538064, 4.76214164818281453252373969762, 5.98543826379842968287378331746, 6.74400648520150631189407193668, 7.08895065610519308635593052039, 8.120799071850939376041909440426

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