Properties

Label 2-6930-1.1-c1-0-19
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 − 6·17-s + 2·19-s − 20-s − 22-s + 6·23-s + 25-s − 2·26-s + 28-s + 8·31-s − 32-s + 6·34-s − 35-s + 8·37-s − 2·38-s + 40-s − 4·43-s + 44-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 − 1.45·17-s + 0.458·19-s − 0.223·20-s − 0.213·22-s + 1.25·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s + 1.43·31-s − 0.176·32-s + 1.02·34-s − 0.169·35-s + 1.31·37-s − 0.324·38-s + 0.158·40-s − 0.609·43-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: 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

\(L(1)\) \(\approx\) \(1.364041629\)
\(L(\frac12)\) \(\approx\) \(1.364041629\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−7.961778104760769718731558399939, −7.45309184490894129165655349329, −6.49440962604456990636621030242, −6.25367743349795758455323227377, −4.94357038776222701645227283953, −4.48854957921016093577088018823, −3.43207112803786454487926086359, −2.66952622753625006218141877321, −1.61465266032745637085546260184, −0.69023203131919387620237309728, 0.69023203131919387620237309728, 1.61465266032745637085546260184, 2.66952622753625006218141877321, 3.43207112803786454487926086359, 4.48854957921016093577088018823, 4.94357038776222701645227283953, 6.25367743349795758455323227377, 6.49440962604456990636621030242, 7.45309184490894129165655349329, 7.961778104760769718731558399939

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