Properties

Label 2-6422-1.1-c1-0-13
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 − 0.445·3-s + 4-s + 0.753·5-s + 0.445·6-s − 1.60·7-s − 8-s − 2.80·9-s − 0.753·10-s − 3.60·11-s − 0.445·12-s + 1.60·14-s − 0.335·15-s + 16-s − 1.40·17-s + 2.80·18-s − 19-s + 0.753·20-s + 0.713·21-s + 3.60·22-s − 8.59·23-s + 0.445·24-s − 4.43·25-s + 2.58·27-s − 1.60·28-s − 1.38·29-s + 0.335·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.256·3-s + 0.5·4-s + 0.336·5-s + 0.181·6-s − 0.606·7-s − 0.353·8-s − 0.933·9-s − 0.238·10-s − 1.08·11-s − 0.128·12-s + 0.428·14-s − 0.0865·15-s + 0.250·16-s − 0.340·17-s + 0.660·18-s − 0.229·19-s + 0.168·20-s + 0.155·21-s + 0.768·22-s − 1.79·23-s + 0.0908·24-s − 0.886·25-s + 0.496·27-s − 0.303·28-s − 0.257·29-s + 0.0611·30-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: $\chi_{6422} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6422,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3891246952\)
\(L(\frac12)\) \(\approx\) \(0.3891246952\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.445T + 3T^{2} \)
5 \( 1 - 0.753T + 5T^{2} \)
7 \( 1 + 1.60T + 7T^{2} \)
11 \( 1 + 3.60T + 11T^{2} \)
17 \( 1 + 1.40T + 17T^{2} \)
23 \( 1 + 8.59T + 23T^{2} \)
29 \( 1 + 1.38T + 29T^{2} \)
31 \( 1 - 2.93T + 31T^{2} \)
37 \( 1 - 1.32T + 37T^{2} \)
41 \( 1 - 2.61T + 41T^{2} \)
43 \( 1 - 1.78T + 43T^{2} \)
47 \( 1 - 0.219T + 47T^{2} \)
53 \( 1 + 5.50T + 53T^{2} \)
59 \( 1 - 1.42T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 + 2.66T + 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 + 1.40T + 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 - 3.10T + 83T^{2} \)
89 \( 1 + 2.19T + 89T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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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

−7.912319902498915708627680453970, −7.62975890065188176774797410668, −6.37620833273903950312954745428, −6.12139456677838306510917710531, −5.42537299189259288773018379559, −4.45528246335055739215434978239, −3.38268243412735518275813664820, −2.58838842449701138410649643717, −1.88472552262529936655715397395, −0.33979266278990368143120891231, 0.33979266278990368143120891231, 1.88472552262529936655715397395, 2.58838842449701138410649643717, 3.38268243412735518275813664820, 4.45528246335055739215434978239, 5.42537299189259288773018379559, 6.12139456677838306510917710531, 6.37620833273903950312954745428, 7.62975890065188176774797410668, 7.912319902498915708627680453970

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