Properties

Label 2-6422-1.1-c1-0-122
Degree $2$
Conductor $6422$
Sign $1$
Analytic cond. $51.2799$
Root an. cond. $7.16100$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s − 2·9-s + 6·11-s + 12-s + 14-s + 16-s + 3·17-s − 2·18-s − 19-s + 21-s + 6·22-s + 3·23-s + 24-s − 5·25-s − 5·27-s + 28-s + 9·29-s + 4·31-s + 32-s + 6·33-s + 3·34-s − 2·36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 1.80·11-s + 0.288·12-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.471·18-s − 0.229·19-s + 0.218·21-s + 1.27·22-s + 0.625·23-s + 0.204·24-s − 25-s − 0.962·27-s + 0.188·28-s + 1.67·29-s + 0.718·31-s + 0.176·32-s + 1.04·33-s + 0.514·34-s − 1/3·36-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: yes
Arithmetic: yes
Character: $\chi_{6422} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6422,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.719303310\)
\(L(\frac12)\) \(\approx\) \(4.719303310\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.969576930662872845166803888368, −7.35558189285866919427797765822, −6.25338153432601213583646709472, −6.18304856110129897953773330946, −5.00748366820747128480894903619, −4.36756058739341503144365761002, −3.55502409354175041648709074357, −2.97707225880527315032523526567, −1.96927933577139967417741574766, −1.05236257727154933297058348414, 1.05236257727154933297058348414, 1.96927933577139967417741574766, 2.97707225880527315032523526567, 3.55502409354175041648709074357, 4.36756058739341503144365761002, 5.00748366820747128480894903619, 6.18304856110129897953773330946, 6.25338153432601213583646709472, 7.35558189285866919427797765822, 7.969576930662872845166803888368

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