Properties

Label 2-6422-1.1-c1-0-14
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 − 1.95·3-s + 4-s − 0.613·5-s + 1.95·6-s − 0.465·7-s − 8-s + 0.829·9-s + 0.613·10-s − 4.22·11-s − 1.95·12-s + 0.465·14-s + 1.20·15-s + 16-s + 5.33·17-s − 0.829·18-s − 19-s − 0.613·20-s + 0.911·21-s + 4.22·22-s − 6.57·23-s + 1.95·24-s − 4.62·25-s + 4.24·27-s − 0.465·28-s + 8.11·29-s − 1.20·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.12·3-s + 0.5·4-s − 0.274·5-s + 0.798·6-s − 0.176·7-s − 0.353·8-s + 0.276·9-s + 0.194·10-s − 1.27·11-s − 0.564·12-s + 0.124·14-s + 0.310·15-s + 0.250·16-s + 1.29·17-s − 0.195·18-s − 0.229·19-s − 0.137·20-s + 0.198·21-s + 0.900·22-s − 1.37·23-s + 0.399·24-s − 0.924·25-s + 0.817·27-s − 0.0880·28-s + 1.50·29-s − 0.219·30-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

\(L(1)\) \(\approx\) \(0.3133638636\)
\(L(\frac12)\) \(\approx\) \(0.3133638636\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.95T + 3T^{2} \)
5 \( 1 + 0.613T + 5T^{2} \)
7 \( 1 + 0.465T + 7T^{2} \)
11 \( 1 + 4.22T + 11T^{2} \)
17 \( 1 - 5.33T + 17T^{2} \)
23 \( 1 + 6.57T + 23T^{2} \)
29 \( 1 - 8.11T + 29T^{2} \)
31 \( 1 - 4.04T + 31T^{2} \)
37 \( 1 + 1.69T + 37T^{2} \)
41 \( 1 + 4.71T + 41T^{2} \)
43 \( 1 - 2.48T + 43T^{2} \)
47 \( 1 + 12.8T + 47T^{2} \)
53 \( 1 + 3.74T + 53T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 + 4.02T + 61T^{2} \)
67 \( 1 + 9.47T + 67T^{2} \)
71 \( 1 - 16.6T + 71T^{2} \)
73 \( 1 - 14.0T + 73T^{2} \)
79 \( 1 + 6.66T + 79T^{2} \)
83 \( 1 + 6.70T + 83T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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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

−8.057411074075015535905290655565, −7.48617212553629809787206005013, −6.36865085175089726798862544683, −6.15635127463988669672125713368, −5.23493032339159060530081852284, −4.69786012209156414272136599264, −3.47184103898271154365743638056, −2.69973989451488256524276051494, −1.53547444577260145763153842093, −0.34500317277520716273995184808, 0.34500317277520716273995184808, 1.53547444577260145763153842093, 2.69973989451488256524276051494, 3.47184103898271154365743638056, 4.69786012209156414272136599264, 5.23493032339159060530081852284, 6.15635127463988669672125713368, 6.36865085175089726798862544683, 7.48617212553629809787206005013, 8.057411074075015535905290655565

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