Properties

Label 2-6422-1.1-c1-0-9
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 − 3.28·3-s + 4-s − 2.88·5-s + 3.28·6-s − 3.02·7-s − 8-s + 7.80·9-s + 2.88·10-s − 2.88·11-s − 3.28·12-s + 3.02·14-s + 9.47·15-s + 16-s + 0.0993·17-s − 7.80·18-s + 19-s − 2.88·20-s + 9.94·21-s + 2.88·22-s + 7.51·23-s + 3.28·24-s + 3.30·25-s − 15.7·27-s − 3.02·28-s + 3.34·29-s − 9.47·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.89·3-s + 0.5·4-s − 1.28·5-s + 1.34·6-s − 1.14·7-s − 0.353·8-s + 2.60·9-s + 0.911·10-s − 0.868·11-s − 0.948·12-s + 0.808·14-s + 2.44·15-s + 0.250·16-s + 0.0240·17-s − 1.83·18-s + 0.229·19-s − 0.644·20-s + 2.16·21-s + 0.614·22-s + 1.56·23-s + 0.670·24-s + 0.661·25-s − 3.03·27-s − 0.571·28-s + 0.620·29-s − 1.72·30-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

\(L(1)\) \(\approx\) \(0.1426497496\)
\(L(\frac12)\) \(\approx\) \(0.1426497496\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.28T + 3T^{2} \)
5 \( 1 + 2.88T + 5T^{2} \)
7 \( 1 + 3.02T + 7T^{2} \)
11 \( 1 + 2.88T + 11T^{2} \)
17 \( 1 - 0.0993T + 17T^{2} \)
23 \( 1 - 7.51T + 23T^{2} \)
29 \( 1 - 3.34T + 29T^{2} \)
31 \( 1 + 5.90T + 31T^{2} \)
37 \( 1 - 6.88T + 37T^{2} \)
41 \( 1 + 6.94T + 41T^{2} \)
43 \( 1 - 6.08T + 43T^{2} \)
47 \( 1 + 5.88T + 47T^{2} \)
53 \( 1 - 9.17T + 53T^{2} \)
59 \( 1 + 0.856T + 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 + 8.97T + 71T^{2} \)
73 \( 1 - 0.646T + 73T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 - 0.951T + 89T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−7.72669246639272191242794982859, −7.23852821049164265069415650323, −6.73517336899600491510974670455, −6.00369130806853447063578704984, −5.28960651287926664784385316679, −4.56702475973427210998385123240, −3.68206145026329823932668471157, −2.80461805608176712481118442454, −1.19897412998178447944003785308, −0.27729597412572189533102691403, 0.27729597412572189533102691403, 1.19897412998178447944003785308, 2.80461805608176712481118442454, 3.68206145026329823932668471157, 4.56702475973427210998385123240, 5.28960651287926664784385316679, 6.00369130806853447063578704984, 6.73517336899600491510974670455, 7.23852821049164265069415650323, 7.72669246639272191242794982859

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