Properties

Label 2-6930-1.1-c1-0-1
Degree $2$
Conductor $6930$
Sign $1$
Analytic cond. $55.3363$
Root an. cond. $7.43883$
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 + 4-s − 5-s − 7-s − 8-s + 10-s − 11-s + 0.941·13-s + 14-s + 16-s − 6.49·17-s − 4.36·19-s − 20-s + 22-s − 6.24·23-s + 25-s − 0.941·26-s − 28-s − 8.74·29-s − 9.55·31-s − 32-s + 6.49·34-s + 35-s + 4.24·37-s + 4.36·38-s + 40-s + 2.13·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.261·13-s + 0.267·14-s + 0.250·16-s − 1.57·17-s − 1.00·19-s − 0.223·20-s + 0.213·22-s − 1.30·23-s + 0.200·25-s − 0.184·26-s − 0.188·28-s − 1.62·29-s − 1.71·31-s − 0.176·32-s + 1.11·34-s + 0.169·35-s + 0.698·37-s + 0.708·38-s + 0.158·40-s + 0.332·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6930\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(55.3363\)
Root analytic conductor: \(7.43883\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6930} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6930,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4718517321\)
\(L(\frac12)\) \(\approx\) \(0.4718517321\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 - 0.941T + 13T^{2} \)
17 \( 1 + 6.49T + 17T^{2} \)
19 \( 1 + 4.36T + 19T^{2} \)
23 \( 1 + 6.24T + 23T^{2} \)
29 \( 1 + 8.74T + 29T^{2} \)
31 \( 1 + 9.55T + 31T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 - 2.13T + 41T^{2} \)
43 \( 1 - 7.67T + 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 - 4.74T + 53T^{2} \)
59 \( 1 - 1.88T + 59T^{2} \)
61 \( 1 - 9.11T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 - 14.6T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 8.36T + 79T^{2} \)
83 \( 1 - 8.49T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−8.027476634706084123685338396581, −7.34053806389158869371996173924, −6.66462126047522891641920631926, −6.04801143499359062044868100117, −5.23430529389800706436179969432, −4.09078616397817139346086260539, −3.72237190359251210183024124206, −2.42548850872596634164798608899, −1.90798638556792342369918824206, −0.36701066668919928526512795685, 0.36701066668919928526512795685, 1.90798638556792342369918824206, 2.42548850872596634164798608899, 3.72237190359251210183024124206, 4.09078616397817139346086260539, 5.23430529389800706436179969432, 6.04801143499359062044868100117, 6.66462126047522891641920631926, 7.34053806389158869371996173924, 8.027476634706084123685338396581

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