Properties

Label 2-6422-1.1-c1-0-220
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 + 1.04·3-s + 4-s + 0.929·5-s + 1.04·6-s − 2.89·7-s + 8-s − 1.90·9-s + 0.929·10-s + 3.97·11-s + 1.04·12-s − 2.89·14-s + 0.972·15-s + 16-s − 4.54·17-s − 1.90·18-s − 19-s + 0.929·20-s − 3.02·21-s + 3.97·22-s − 4.09·23-s + 1.04·24-s − 4.13·25-s − 5.13·27-s − 2.89·28-s + 1.64·29-s + 0.972·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.604·3-s + 0.5·4-s + 0.415·5-s + 0.427·6-s − 1.09·7-s + 0.353·8-s − 0.635·9-s + 0.293·10-s + 1.19·11-s + 0.302·12-s − 0.773·14-s + 0.251·15-s + 0.250·16-s − 1.10·17-s − 0.449·18-s − 0.229·19-s + 0.207·20-s − 0.660·21-s + 0.847·22-s − 0.854·23-s + 0.213·24-s − 0.827·25-s − 0.987·27-s − 0.546·28-s + 0.306·29-s + 0.177·30-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 \( 1 - 1.04T + 3T^{2} \)
5 \( 1 - 0.929T + 5T^{2} \)
7 \( 1 + 2.89T + 7T^{2} \)
11 \( 1 - 3.97T + 11T^{2} \)
17 \( 1 + 4.54T + 17T^{2} \)
23 \( 1 + 4.09T + 23T^{2} \)
29 \( 1 - 1.64T + 29T^{2} \)
31 \( 1 + 7.79T + 31T^{2} \)
37 \( 1 - 2.85T + 37T^{2} \)
41 \( 1 + 0.453T + 41T^{2} \)
43 \( 1 - 4.16T + 43T^{2} \)
47 \( 1 + 9.08T + 47T^{2} \)
53 \( 1 + 3.63T + 53T^{2} \)
59 \( 1 - 9.02T + 59T^{2} \)
61 \( 1 - 4.87T + 61T^{2} \)
67 \( 1 + 14.2T + 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 - 9.60T + 79T^{2} \)
83 \( 1 + 4.33T + 83T^{2} \)
89 \( 1 - 6.23T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−7.52855106537430001609690574463, −6.79667044500320180745151667589, −6.08893873981135978506799866183, −5.84397132911193303288259677100, −4.62005592786168085316181686486, −3.85698608871129571866448415000, −3.31923270439335011526052560897, −2.43153593833644305115230824614, −1.71901895879406836595036892106, 0, 1.71901895879406836595036892106, 2.43153593833644305115230824614, 3.31923270439335011526052560897, 3.85698608871129571866448415000, 4.62005592786168085316181686486, 5.84397132911193303288259677100, 6.08893873981135978506799866183, 6.79667044500320180745151667589, 7.52855106537430001609690574463

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