Properties

Label 2-6930-1.1-c1-0-33
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 + 3.46·13-s − 14-s + 16-s + 3.46·17-s + 20-s + 22-s + 1.46·23-s + 25-s − 3.46·26-s + 28-s − 2·29-s + 6.92·31-s − 32-s − 3.46·34-s + 35-s + 7.46·37-s − 40-s + 8.92·41-s − 2.92·43-s − 44-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 + 0.960·13-s − 0.267·14-s + 0.250·16-s + 0.840·17-s + 0.223·20-s + 0.213·22-s + 0.305·23-s + 0.200·25-s − 0.679·26-s + 0.188·28-s − 0.371·29-s + 1.24·31-s − 0.176·32-s − 0.594·34-s + 0.169·35-s + 1.22·37-s − 0.158·40-s + 1.39·41-s − 0.446·43-s − 0.150·44-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

\(L(1)\) \(\approx\) \(1.871902238\)
\(L(\frac12)\) \(\approx\) \(1.871902238\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 1.46T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 - 7.46T + 37T^{2} \)
41 \( 1 - 8.92T + 41T^{2} \)
43 \( 1 + 2.92T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 - 2.92T + 59T^{2} \)
61 \( 1 + 4.92T + 61T^{2} \)
67 \( 1 - 9.46T + 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 0.928T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 6.92T + 83T^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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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

−8.069010067391974921787754007440, −7.45528631651160523902631569633, −6.49695804129831592607330074785, −6.02978089617071476213042676222, −5.24807301524678661547440328364, −4.40666001660186538206808704401, −3.37816480894177620080557535980, −2.61843441706373356850271154758, −1.60998436757552426509478931467, −0.829829878682742375016200558201, 0.829829878682742375016200558201, 1.60998436757552426509478931467, 2.61843441706373356850271154758, 3.37816480894177620080557535980, 4.40666001660186538206808704401, 5.24807301524678661547440328364, 6.02978089617071476213042676222, 6.49695804129831592607330074785, 7.45528631651160523902631569633, 8.069010067391974921787754007440

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