Properties

Label 2-6422-1.1-c1-0-84
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.60·3-s + 4-s − 0.883·5-s + 2.60·6-s − 3.97·7-s − 8-s + 3.77·9-s + 0.883·10-s + 4.30·11-s − 2.60·12-s + 3.97·14-s + 2.29·15-s + 16-s + 2.64·17-s − 3.77·18-s + 19-s − 0.883·20-s + 10.3·21-s − 4.30·22-s − 3.14·23-s + 2.60·24-s − 4.22·25-s − 2.00·27-s − 3.97·28-s − 3.42·29-s − 2.29·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.50·3-s + 0.5·4-s − 0.394·5-s + 1.06·6-s − 1.50·7-s − 0.353·8-s + 1.25·9-s + 0.279·10-s + 1.29·11-s − 0.751·12-s + 1.06·14-s + 0.593·15-s + 0.250·16-s + 0.642·17-s − 0.888·18-s + 0.229·19-s − 0.197·20-s + 2.25·21-s − 0.918·22-s − 0.655·23-s + 0.531·24-s − 0.844·25-s − 0.386·27-s − 0.751·28-s − 0.636·29-s − 0.419·30-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: \(1\)
Selberg data: \((2,\ 6422,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.60T + 3T^{2} \)
5 \( 1 + 0.883T + 5T^{2} \)
7 \( 1 + 3.97T + 7T^{2} \)
11 \( 1 - 4.30T + 11T^{2} \)
17 \( 1 - 2.64T + 17T^{2} \)
23 \( 1 + 3.14T + 23T^{2} \)
29 \( 1 + 3.42T + 29T^{2} \)
31 \( 1 + 7.08T + 31T^{2} \)
37 \( 1 + 2.46T + 37T^{2} \)
41 \( 1 + 2.17T + 41T^{2} \)
43 \( 1 - 9.90T + 43T^{2} \)
47 \( 1 + 9.90T + 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 - 5.24T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 - 8.80T + 67T^{2} \)
71 \( 1 - 1.84T + 71T^{2} \)
73 \( 1 + 7.73T + 73T^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 - 8.61T + 83T^{2} \)
89 \( 1 - 7.36T + 89T^{2} \)
97 \( 1 + 3.40T + 97T^{2} \)
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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

−7.41655312315471876846453640887, −6.92688102600557335320905814993, −6.17383561741646790858037655651, −5.94214239743765829330634402696, −4.99578052456473153920622882661, −3.78944690566711099525626068209, −3.48004027688468917670641333444, −1.97611466342774675151386757230, −0.842140471235616929538368583969, 0, 0.842140471235616929538368583969, 1.97611466342774675151386757230, 3.48004027688468917670641333444, 3.78944690566711099525626068209, 4.99578052456473153920622882661, 5.94214239743765829330634402696, 6.17383561741646790858037655651, 6.92688102600557335320905814993, 7.41655312315471876846453640887

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