Properties

Label 2-6930-1.1-c1-0-4
Degree $2$
Conductor $6930$
Sign $1$
Analytic cond. $55.3363$
Root an. cond. $7.43883$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 2·13-s − 14-s + 16-s − 2·17-s − 4·19-s − 20-s − 22-s − 8·23-s + 25-s + 2·26-s + 28-s − 6·29-s + 8·31-s − 32-s + 2·34-s − 35-s − 2·37-s + 4·38-s + 40-s − 2·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.223·20-s − 0.213·22-s − 1.66·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s − 0.169·35-s − 0.328·37-s + 0.648·38-s + 0.158·40-s − 0.312·41-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: yes
Arithmetic: yes
Character: $\chi_{6930} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6930,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9075788965\)
\(L(\frac12)\) \(\approx\) \(0.9075788965\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.973713173933110947926764864448, −7.45935873328665285211014998706, −6.62124918534133464749967251506, −6.08664865510948583917892553092, −5.10046287983341572633808332872, −4.28569018834161615989680894313, −3.62964123203857338701229904242, −2.43690865935129462013858630439, −1.83614978588270814499130694422, −0.52531453803820092568863048296, 0.52531453803820092568863048296, 1.83614978588270814499130694422, 2.43690865935129462013858630439, 3.62964123203857338701229904242, 4.28569018834161615989680894313, 5.10046287983341572633808332872, 6.08664865510948583917892553092, 6.62124918534133464749967251506, 7.45935873328665285211014998706, 7.973713173933110947926764864448

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