Properties

Label 2-6930-1.1-c1-0-11
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 + 4.68·13-s + 14-s + 16-s + 0.292·17-s + 6.51·19-s − 20-s + 22-s − 2.85·23-s + 25-s − 4.68·26-s − 28-s + 1.43·29-s + 0.978·31-s − 32-s − 0.292·34-s + 35-s + 0.853·37-s − 6.51·38-s + 40-s + 6.22·41-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 + 1.29·13-s + 0.267·14-s + 0.250·16-s + 0.0709·17-s + 1.49·19-s − 0.223·20-s + 0.213·22-s − 0.595·23-s + 0.200·25-s − 0.918·26-s − 0.188·28-s + 0.267·29-s + 0.175·31-s − 0.176·32-s − 0.0502·34-s + 0.169·35-s + 0.140·37-s − 1.05·38-s + 0.158·40-s + 0.972·41-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

\(L(1)\) \(\approx\) \(1.242337311\)
\(L(\frac12)\) \(\approx\) \(1.242337311\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.68T + 13T^{2} \)
17 \( 1 - 0.292T + 17T^{2} \)
19 \( 1 - 6.51T + 19T^{2} \)
23 \( 1 + 2.85T + 23T^{2} \)
29 \( 1 - 1.43T + 29T^{2} \)
31 \( 1 - 0.978T + 31T^{2} \)
37 \( 1 - 0.853T + 37T^{2} \)
41 \( 1 - 6.22T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 - 9.95T + 47T^{2} \)
53 \( 1 + 5.43T + 53T^{2} \)
59 \( 1 - 9.37T + 59T^{2} \)
61 \( 1 + 11.9T + 61T^{2} \)
67 \( 1 + 0.585T + 67T^{2} \)
71 \( 1 - 0.335T + 71T^{2} \)
73 \( 1 + 3.70T + 73T^{2} \)
79 \( 1 + 2.51T + 79T^{2} \)
83 \( 1 - 1.70T + 83T^{2} \)
89 \( 1 + 13.0T + 89T^{2} \)
97 \( 1 - 9.10T + 97T^{2} \)
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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

−7.900541596082541343454508279587, −7.48060808125544692999137989541, −6.62899616370167223782100440452, −5.98922604176304663676044977178, −5.28383837323806687741950422588, −4.21641061333996531826932461763, −3.43264114714875048809737981738, −2.78706925799825349980546117231, −1.56751638773802829439853075035, −0.66264159211278378958030835745, 0.66264159211278378958030835745, 1.56751638773802829439853075035, 2.78706925799825349980546117231, 3.43264114714875048809737981738, 4.21641061333996531826932461763, 5.28383837323806687741950422588, 5.98922604176304663676044977178, 6.62899616370167223782100440452, 7.48060808125544692999137989541, 7.900541596082541343454508279587

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