Properties

Label 2-6422-1.1-c1-0-130
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 + 3.24·3-s + 4-s − 1.55·5-s − 3.24·6-s + 3.60·7-s − 8-s + 7.54·9-s + 1.55·10-s + 5.18·11-s + 3.24·12-s − 3.60·14-s − 5.04·15-s + 16-s − 5.96·17-s − 7.54·18-s + 19-s − 1.55·20-s + 11.7·21-s − 5.18·22-s + 4.29·23-s − 3.24·24-s − 2.58·25-s + 14.7·27-s + 3.60·28-s + 2.91·29-s + 5.04·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.87·3-s + 0.5·4-s − 0.695·5-s − 1.32·6-s + 1.36·7-s − 0.353·8-s + 2.51·9-s + 0.491·10-s + 1.56·11-s + 0.937·12-s − 0.963·14-s − 1.30·15-s + 0.250·16-s − 1.44·17-s − 1.77·18-s + 0.229·19-s − 0.347·20-s + 2.55·21-s − 1.10·22-s + 0.895·23-s − 0.662·24-s − 0.516·25-s + 2.83·27-s + 0.681·28-s + 0.540·29-s + 0.921·30-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

\(L(1)\) \(\approx\) \(3.664156741\)
\(L(\frac12)\) \(\approx\) \(3.664156741\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.24T + 3T^{2} \)
5 \( 1 + 1.55T + 5T^{2} \)
7 \( 1 - 3.60T + 7T^{2} \)
11 \( 1 - 5.18T + 11T^{2} \)
17 \( 1 + 5.96T + 17T^{2} \)
23 \( 1 - 4.29T + 23T^{2} \)
29 \( 1 - 2.91T + 29T^{2} \)
31 \( 1 + 6.04T + 31T^{2} \)
37 \( 1 - 6.71T + 37T^{2} \)
41 \( 1 - 9.83T + 41T^{2} \)
43 \( 1 - 5.16T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 - 3.82T + 53T^{2} \)
59 \( 1 + 13.5T + 59T^{2} \)
61 \( 1 + 0.615T + 61T^{2} \)
67 \( 1 - 0.987T + 67T^{2} \)
71 \( 1 - 4.71T + 71T^{2} \)
73 \( 1 + 6.59T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 3.91T + 83T^{2} \)
89 \( 1 + 0.789T + 89T^{2} \)
97 \( 1 - 15.9T + 97T^{2} \)
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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

−8.048644197633118179010055267359, −7.64059758885608525410341320378, −7.04580436118941143151737382603, −6.26840291579101304657555160246, −4.73428507128285680828125327435, −4.22112406399335145458596198992, −3.56996303844386388672325246102, −2.58370261047561841004407938892, −1.82508024498627528078823083091, −1.11963269246655813783023111526, 1.11963269246655813783023111526, 1.82508024498627528078823083091, 2.58370261047561841004407938892, 3.56996303844386388672325246102, 4.22112406399335145458596198992, 4.73428507128285680828125327435, 6.26840291579101304657555160246, 7.04580436118941143151737382603, 7.64059758885608525410341320378, 8.048644197633118179010055267359

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