Properties

Label 2-6422-1.1-c1-0-22
Degree $2$
Conductor $6422$
Sign $1$
Analytic cond. $51.2799$
Root an. cond. $7.16100$
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 − 2.64·3-s + 4-s + 3.69·5-s + 2.64·6-s − 4.07·7-s − 8-s + 3.99·9-s − 3.69·10-s − 5.71·11-s − 2.64·12-s + 4.07·14-s − 9.76·15-s + 16-s + 3.36·17-s − 3.99·18-s − 19-s + 3.69·20-s + 10.7·21-s + 5.71·22-s + 1.44·23-s + 2.64·24-s + 8.63·25-s − 2.63·27-s − 4.07·28-s + 3.22·29-s + 9.76·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.52·3-s + 0.5·4-s + 1.65·5-s + 1.07·6-s − 1.53·7-s − 0.353·8-s + 1.33·9-s − 1.16·10-s − 1.72·11-s − 0.763·12-s + 1.08·14-s − 2.52·15-s + 0.250·16-s + 0.816·17-s − 0.942·18-s − 0.229·19-s + 0.825·20-s + 2.35·21-s + 1.21·22-s + 0.301·23-s + 0.539·24-s + 1.72·25-s − 0.508·27-s − 0.769·28-s + 0.599·29-s + 1.78·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6422\)    =    \(2 \cdot 13^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(51.2799\)
Root analytic conductor: \(7.16100\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6422,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4498076716\)
\(L(\frac12)\) \(\approx\) \(0.4498076716\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
13 \( 1 \)
19 \( 1 + T \)
good3 \( 1 + 2.64T + 3T^{2} \)
5 \( 1 - 3.69T + 5T^{2} \)
7 \( 1 + 4.07T + 7T^{2} \)
11 \( 1 + 5.71T + 11T^{2} \)
17 \( 1 - 3.36T + 17T^{2} \)
23 \( 1 - 1.44T + 23T^{2} \)
29 \( 1 - 3.22T + 29T^{2} \)
31 \( 1 + 5.87T + 31T^{2} \)
37 \( 1 + 11.7T + 37T^{2} \)
41 \( 1 + 1.52T + 41T^{2} \)
43 \( 1 - 1.05T + 43T^{2} \)
47 \( 1 - 3.95T + 47T^{2} \)
53 \( 1 + 7.94T + 53T^{2} \)
59 \( 1 - 1.81T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 + 7.39T + 71T^{2} \)
73 \( 1 + 1.81T + 73T^{2} \)
79 \( 1 - 4.47T + 79T^{2} \)
83 \( 1 - 7.57T + 83T^{2} \)
89 \( 1 - 4.91T + 89T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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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

−7.936641963122060252350203147590, −6.97587076535837561930287361068, −6.60031573067667712445947324301, −5.81144891860919186042798076688, −5.54654030188770693806939798211, −4.90556483005522048600924125759, −3.32795378637141851612742728439, −2.60027437209605746233416439488, −1.59757209672983397690166097632, −0.41289113158626736094046493470, 0.41289113158626736094046493470, 1.59757209672983397690166097632, 2.60027437209605746233416439488, 3.32795378637141851612742728439, 4.90556483005522048600924125759, 5.54654030188770693806939798211, 5.81144891860919186042798076688, 6.60031573067667712445947324301, 6.97587076535837561930287361068, 7.936641963122060252350203147590

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