Properties

Label 2-6762-1.1-c1-0-103
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 + 0.585·5-s + 6-s + 8-s + 9-s + 0.585·10-s + 5.93·11-s + 12-s + 6.45·13-s + 0.585·15-s + 16-s − 0.371·17-s + 18-s + 3.26·19-s + 0.585·20-s + 5.93·22-s + 23-s + 24-s − 4.65·25-s + 6.45·26-s + 27-s − 0.948·29-s + 0.585·30-s + 1.26·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.261·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.185·10-s + 1.78·11-s + 0.288·12-s + 1.79·13-s + 0.151·15-s + 0.250·16-s − 0.0901·17-s + 0.235·18-s + 0.747·19-s + 0.130·20-s + 1.26·22-s + 0.208·23-s + 0.204·24-s − 0.931·25-s + 1.26·26-s + 0.192·27-s − 0.176·29-s + 0.106·30-s + 0.226·31-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

\(L(1)\) \(\approx\) \(5.565268023\)
\(L(\frac12)\) \(\approx\) \(5.565268023\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.585T + 5T^{2} \)
11 \( 1 - 5.93T + 11T^{2} \)
13 \( 1 - 6.45T + 13T^{2} \)
17 \( 1 + 0.371T + 17T^{2} \)
19 \( 1 - 3.26T + 19T^{2} \)
29 \( 1 + 0.948T + 29T^{2} \)
31 \( 1 - 1.26T + 31T^{2} \)
37 \( 1 - 0.371T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 + 6.54T + 43T^{2} \)
47 \( 1 - 10.6T + 47T^{2} \)
53 \( 1 + 8.23T + 53T^{2} \)
59 \( 1 - 1.05T + 59T^{2} \)
61 \( 1 + 4.44T + 61T^{2} \)
67 \( 1 + 0.0286T + 67T^{2} \)
71 \( 1 - 2.38T + 71T^{2} \)
73 \( 1 - 4.76T + 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 - 2.51T + 83T^{2} \)
89 \( 1 + 15.2T + 89T^{2} \)
97 \( 1 - 8.85T + 97T^{2} \)
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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

−8.004004790914866377383081454515, −7.09161220095080563302477279989, −6.46163648158524132527111341047, −5.99248073775134266337715682523, −5.11496613284156610403570399138, −4.05647369741984547701059810062, −3.73401829890525961926494350287, −2.99713995526637507318493328235, −1.73833058124905428566825319584, −1.23936912562065374082258074901, 1.23936912562065374082258074901, 1.73833058124905428566825319584, 2.99713995526637507318493328235, 3.73401829890525961926494350287, 4.05647369741984547701059810062, 5.11496613284156610403570399138, 5.99248073775134266337715682523, 6.46163648158524132527111341047, 7.09161220095080563302477279989, 8.004004790914866377383081454515

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