Properties

Label 2-6930-1.1-c1-0-38
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 − 0.694·13-s − 14-s + 16-s + 2.69·17-s + 5.51·19-s + 20-s − 22-s + 7.43·23-s + 25-s + 0.694·26-s + 28-s + 0.610·29-s − 7.51·31-s − 32-s − 2.69·34-s + 35-s − 1.30·37-s − 5.51·38-s − 40-s + 9.51·41-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 − 0.192·13-s − 0.267·14-s + 0.250·16-s + 0.653·17-s + 1.26·19-s + 0.223·20-s − 0.213·22-s + 1.55·23-s + 0.200·25-s + 0.136·26-s + 0.188·28-s + 0.113·29-s − 1.35·31-s − 0.176·32-s − 0.462·34-s + 0.169·35-s − 0.214·37-s − 0.895·38-s − 0.158·40-s + 1.48·41-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6930,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.923540698\)
\(L(\frac12)\) \(\approx\) \(1.923540698\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.694T + 13T^{2} \)
17 \( 1 - 2.69T + 17T^{2} \)
19 \( 1 - 5.51T + 19T^{2} \)
23 \( 1 - 7.43T + 23T^{2} \)
29 \( 1 - 0.610T + 29T^{2} \)
31 \( 1 + 7.51T + 31T^{2} \)
37 \( 1 + 1.30T + 37T^{2} \)
41 \( 1 - 9.51T + 41T^{2} \)
43 \( 1 - 5.51T + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + 4.12T + 53T^{2} \)
59 \( 1 + 3.51T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + 8.95T + 67T^{2} \)
71 \( 1 - 4.77T + 71T^{2} \)
73 \( 1 - 5.51T + 73T^{2} \)
79 \( 1 + 2.77T + 79T^{2} \)
83 \( 1 + 4.73T + 83T^{2} \)
89 \( 1 - 8.04T + 89T^{2} \)
97 \( 1 + 6.90T + 97T^{2} \)
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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

−7.83137692836985750234563017543, −7.39421500286946052716792106937, −6.77158758952053823195426772354, −5.75988713873773532899181444454, −5.36724751294884522783624321661, −4.40977347241790630803480861853, −3.34767078732276831058188571422, −2.65085672423893477198451842824, −1.58145068022609358610088140695, −0.851286187519780566946491845247, 0.851286187519780566946491845247, 1.58145068022609358610088140695, 2.65085672423893477198451842824, 3.34767078732276831058188571422, 4.40977347241790630803480861853, 5.36724751294884522783624321661, 5.75988713873773532899181444454, 6.77158758952053823195426772354, 7.39421500286946052716792106937, 7.83137692836985750234563017543

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