Properties

Label 2-6422-1.1-c1-0-118
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3.31·3-s + 4-s − 2.06·5-s + 3.31·6-s − 3.81·7-s + 8-s + 7.96·9-s − 2.06·10-s + 1.60·11-s + 3.31·12-s − 3.81·14-s − 6.85·15-s + 16-s + 7.51·17-s + 7.96·18-s + 19-s − 2.06·20-s − 12.6·21-s + 1.60·22-s + 2.12·23-s + 3.31·24-s − 0.719·25-s + 16.4·27-s − 3.81·28-s − 7.07·29-s − 6.85·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.91·3-s + 0.5·4-s − 0.925·5-s + 1.35·6-s − 1.44·7-s + 0.353·8-s + 2.65·9-s − 0.654·10-s + 0.484·11-s + 0.955·12-s − 1.02·14-s − 1.76·15-s + 0.250·16-s + 1.82·17-s + 1.87·18-s + 0.229·19-s − 0.462·20-s − 2.75·21-s + 0.342·22-s + 0.442·23-s + 0.675·24-s − 0.143·25-s + 3.16·27-s − 0.721·28-s − 1.31·29-s − 1.25·30-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

\(L(1)\) \(\approx\) \(5.498922892\)
\(L(\frac12)\) \(\approx\) \(5.498922892\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.31T + 3T^{2} \)
5 \( 1 + 2.06T + 5T^{2} \)
7 \( 1 + 3.81T + 7T^{2} \)
11 \( 1 - 1.60T + 11T^{2} \)
17 \( 1 - 7.51T + 17T^{2} \)
23 \( 1 - 2.12T + 23T^{2} \)
29 \( 1 + 7.07T + 29T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 - 3.61T + 37T^{2} \)
41 \( 1 + 1.62T + 41T^{2} \)
43 \( 1 - 0.236T + 43T^{2} \)
47 \( 1 + 9.28T + 47T^{2} \)
53 \( 1 + 1.14T + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 - 2.97T + 61T^{2} \)
67 \( 1 + 5.51T + 67T^{2} \)
71 \( 1 - 2.43T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 - 9.88T + 83T^{2} \)
89 \( 1 - 8.37T + 89T^{2} \)
97 \( 1 - 0.865T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.920186889233306452205301184540, −7.43146364504407561187215321653, −6.81433501660292170857948721530, −6.02278073372335120174368362015, −4.88203163282509598587409495189, −3.93600151098975687268223820704, −3.44650618691759367171548860124, −3.20577449498229238405295108441, −2.24941068701778727502933135978, −1.03680704775940595712036422388, 1.03680704775940595712036422388, 2.24941068701778727502933135978, 3.20577449498229238405295108441, 3.44650618691759367171548860124, 3.93600151098975687268223820704, 4.88203163282509598587409495189, 6.02278073372335120174368362015, 6.81433501660292170857948721530, 7.43146364504407561187215321653, 7.920186889233306452205301184540

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