Properties

Label 2-6422-1.1-c1-0-137
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 + 2.47·3-s + 4-s + 0.657·5-s + 2.47·6-s − 1.67·7-s + 8-s + 3.11·9-s + 0.657·10-s + 4.79·11-s + 2.47·12-s − 1.67·14-s + 1.62·15-s + 16-s − 1.91·17-s + 3.11·18-s + 19-s + 0.657·20-s − 4.12·21-s + 4.79·22-s + 4.96·23-s + 2.47·24-s − 4.56·25-s + 0.273·27-s − 1.67·28-s + 6.18·29-s + 1.62·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.42·3-s + 0.5·4-s + 0.294·5-s + 1.00·6-s − 0.631·7-s + 0.353·8-s + 1.03·9-s + 0.207·10-s + 1.44·11-s + 0.713·12-s − 0.446·14-s + 0.419·15-s + 0.250·16-s − 0.465·17-s + 0.733·18-s + 0.229·19-s + 0.147·20-s − 0.900·21-s + 1.02·22-s + 1.03·23-s + 0.504·24-s − 0.913·25-s + 0.0525·27-s − 0.315·28-s + 1.14·29-s + 0.296·30-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

\(L(1)\) \(\approx\) \(6.136517288\)
\(L(\frac12)\) \(\approx\) \(6.136517288\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.47T + 3T^{2} \)
5 \( 1 - 0.657T + 5T^{2} \)
7 \( 1 + 1.67T + 7T^{2} \)
11 \( 1 - 4.79T + 11T^{2} \)
17 \( 1 + 1.91T + 17T^{2} \)
23 \( 1 - 4.96T + 23T^{2} \)
29 \( 1 - 6.18T + 29T^{2} \)
31 \( 1 - 5.42T + 31T^{2} \)
37 \( 1 + 0.670T + 37T^{2} \)
41 \( 1 - 5.58T + 41T^{2} \)
43 \( 1 + 7.81T + 43T^{2} \)
47 \( 1 - 1.70T + 47T^{2} \)
53 \( 1 - 4.47T + 53T^{2} \)
59 \( 1 - 1.28T + 59T^{2} \)
61 \( 1 - 0.0985T + 61T^{2} \)
67 \( 1 - 9.67T + 67T^{2} \)
71 \( 1 + 8.55T + 71T^{2} \)
73 \( 1 + 5.77T + 73T^{2} \)
79 \( 1 - 7.95T + 79T^{2} \)
83 \( 1 - 1.72T + 83T^{2} \)
89 \( 1 - 9.87T + 89T^{2} \)
97 \( 1 - 9.59T + 97T^{2} \)
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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

−8.077744169538190993420111518438, −7.21487239570442246810282213683, −6.58906267761408992616398544854, −6.09239874243517248950786504225, −4.97431098211210237939501235565, −4.16748586296761855389805165233, −3.54198835975428691523562249034, −2.90247031701052661407702350210, −2.15416365606706705669117637708, −1.16321342482419311914367702722, 1.16321342482419311914367702722, 2.15416365606706705669117637708, 2.90247031701052661407702350210, 3.54198835975428691523562249034, 4.16748586296761855389805165233, 4.97431098211210237939501235565, 6.09239874243517248950786504225, 6.58906267761408992616398544854, 7.21487239570442246810282213683, 8.077744169538190993420111518438

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