Properties

Label 2-6422-1.1-c1-0-54
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 + 1.28·3-s + 4-s − 4.29·5-s − 1.28·6-s + 4.70·7-s − 8-s − 1.34·9-s + 4.29·10-s + 3.40·11-s + 1.28·12-s − 4.70·14-s − 5.53·15-s + 16-s − 2.91·17-s + 1.34·18-s + 19-s − 4.29·20-s + 6.05·21-s − 3.40·22-s − 5.88·23-s − 1.28·24-s + 13.4·25-s − 5.58·27-s + 4.70·28-s − 6.28·29-s + 5.53·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.742·3-s + 0.5·4-s − 1.92·5-s − 0.525·6-s + 1.77·7-s − 0.353·8-s − 0.448·9-s + 1.35·10-s + 1.02·11-s + 0.371·12-s − 1.25·14-s − 1.42·15-s + 0.250·16-s − 0.708·17-s + 0.316·18-s + 0.229·19-s − 0.961·20-s + 1.32·21-s − 0.725·22-s − 1.22·23-s − 0.262·24-s + 2.69·25-s − 1.07·27-s + 0.889·28-s − 1.16·29-s + 1.00·30-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

\(L(1)\) \(\approx\) \(1.402294863\)
\(L(\frac12)\) \(\approx\) \(1.402294863\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.28T + 3T^{2} \)
5 \( 1 + 4.29T + 5T^{2} \)
7 \( 1 - 4.70T + 7T^{2} \)
11 \( 1 - 3.40T + 11T^{2} \)
17 \( 1 + 2.91T + 17T^{2} \)
23 \( 1 + 5.88T + 23T^{2} \)
29 \( 1 + 6.28T + 29T^{2} \)
31 \( 1 - 4.34T + 31T^{2} \)
37 \( 1 - 9.73T + 37T^{2} \)
41 \( 1 - 1.32T + 41T^{2} \)
43 \( 1 + 8.24T + 43T^{2} \)
47 \( 1 - 1.55T + 47T^{2} \)
53 \( 1 - 8.55T + 53T^{2} \)
59 \( 1 - 6.50T + 59T^{2} \)
61 \( 1 - 6.39T + 61T^{2} \)
67 \( 1 + 1.20T + 67T^{2} \)
71 \( 1 + 7.74T + 71T^{2} \)
73 \( 1 - 0.805T + 73T^{2} \)
79 \( 1 - 1.33T + 79T^{2} \)
83 \( 1 - 5.24T + 83T^{2} \)
89 \( 1 + 9.68T + 89T^{2} \)
97 \( 1 + 6.38T + 97T^{2} \)
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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

−8.112676373887507755494779112231, −7.61898412615053658304481704089, −7.08673236344663687498210506190, −6.03719053130495471590052519877, −4.95234465982787922102337043585, −4.12135459399540327359417274605, −3.78091681969431399336336923184, −2.65354044510241021977351231007, −1.76244368266079034480042945223, −0.66792143329379907432758296821, 0.66792143329379907432758296821, 1.76244368266079034480042945223, 2.65354044510241021977351231007, 3.78091681969431399336336923184, 4.12135459399540327359417274605, 4.95234465982787922102337043585, 6.03719053130495471590052519877, 7.08673236344663687498210506190, 7.61898412615053658304481704089, 8.112676373887507755494779112231

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