Properties

Label 2-6422-1.1-c1-0-65
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 − 0.953·3-s + 4-s + 3.75·5-s + 0.953·6-s + 3.81·7-s − 8-s − 2.08·9-s − 3.75·10-s − 5.51·11-s − 0.953·12-s − 3.81·14-s − 3.58·15-s + 16-s + 4.01·17-s + 2.08·18-s + 19-s + 3.75·20-s − 3.63·21-s + 5.51·22-s − 8.48·23-s + 0.953·24-s + 9.09·25-s + 4.85·27-s + 3.81·28-s + 3.65·29-s + 3.58·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.550·3-s + 0.5·4-s + 1.67·5-s + 0.389·6-s + 1.44·7-s − 0.353·8-s − 0.696·9-s − 1.18·10-s − 1.66·11-s − 0.275·12-s − 1.01·14-s − 0.924·15-s + 0.250·16-s + 0.973·17-s + 0.492·18-s + 0.229·19-s + 0.839·20-s − 0.793·21-s + 1.17·22-s − 1.76·23-s + 0.194·24-s + 1.81·25-s + 0.934·27-s + 0.720·28-s + 0.677·29-s + 0.653·30-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

\(L(1)\) \(\approx\) \(1.589728129\)
\(L(\frac12)\) \(\approx\) \(1.589728129\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.953T + 3T^{2} \)
5 \( 1 - 3.75T + 5T^{2} \)
7 \( 1 - 3.81T + 7T^{2} \)
11 \( 1 + 5.51T + 11T^{2} \)
17 \( 1 - 4.01T + 17T^{2} \)
23 \( 1 + 8.48T + 23T^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 + 10.5T + 31T^{2} \)
37 \( 1 + 0.902T + 37T^{2} \)
41 \( 1 - 4.14T + 41T^{2} \)
43 \( 1 + 5.76T + 43T^{2} \)
47 \( 1 + 6.37T + 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 - 0.808T + 59T^{2} \)
61 \( 1 + 1.66T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 - 5.59T + 71T^{2} \)
73 \( 1 - 2.82T + 73T^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 - 11.4T + 97T^{2} \)
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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

−8.024883918828963195799904267382, −7.57784534906137404810670338842, −6.47443648221503135166603346004, −5.76277421765113300729264116729, −5.33159230194977636390637927884, −4.94144038845257606285580678765, −3.36385016507737614254103074156, −2.18114246982512537707698954698, −2.00261558319115123779769504560, −0.73699684317271086378026760037, 0.73699684317271086378026760037, 2.00261558319115123779769504560, 2.18114246982512537707698954698, 3.36385016507737614254103074156, 4.94144038845257606285580678765, 5.33159230194977636390637927884, 5.76277421765113300729264116729, 6.47443648221503135166603346004, 7.57784534906137404810670338842, 8.024883918828963195799904267382

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