Properties

Label 2-6930-1.1-c1-0-84
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 $1$

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 − 3.06·13-s + 14-s + 16-s − 5.06·17-s − 3.38·19-s − 20-s − 22-s + 8.58·23-s + 25-s − 3.06·26-s + 28-s + 4.12·29-s + 1.38·31-s + 32-s − 5.06·34-s − 35-s + 1.06·37-s − 3.38·38-s − 40-s − 0.610·41-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 − 0.849·13-s + 0.267·14-s + 0.250·16-s − 1.22·17-s − 0.777·19-s − 0.223·20-s − 0.213·22-s + 1.78·23-s + 0.200·25-s − 0.600·26-s + 0.188·28-s + 0.766·29-s + 0.249·31-s + 0.176·32-s − 0.868·34-s − 0.169·35-s + 0.174·37-s − 0.549·38-s − 0.158·40-s − 0.0953·41-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: \(1\)
Selberg data: \((2,\ 6930,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.06T + 13T^{2} \)
17 \( 1 + 5.06T + 17T^{2} \)
19 \( 1 + 3.38T + 19T^{2} \)
23 \( 1 - 8.58T + 23T^{2} \)
29 \( 1 - 4.12T + 29T^{2} \)
31 \( 1 - 1.38T + 31T^{2} \)
37 \( 1 - 1.06T + 37T^{2} \)
41 \( 1 + 0.610T + 41T^{2} \)
43 \( 1 + 3.38T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 9.51T + 53T^{2} \)
59 \( 1 + 5.38T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 - 15.9T + 67T^{2} \)
71 \( 1 + 14.2T + 71T^{2} \)
73 \( 1 + 3.38T + 73T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + 2.73T + 97T^{2} \)
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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

−7.46058247369271883884266781880, −6.83819790542274522153517022095, −6.30561793755052304715127144252, −5.20482311824732054480007586886, −4.72969424290869985291037188394, −4.20882470592292020496528421962, −3.08366971059549318394839482478, −2.53230337078197413757607874568, −1.46180583769146282861450246261, 0, 1.46180583769146282861450246261, 2.53230337078197413757607874568, 3.08366971059549318394839482478, 4.20882470592292020496528421962, 4.72969424290869985291037188394, 5.20482311824732054480007586886, 6.30561793755052304715127144252, 6.83819790542274522153517022095, 7.46058247369271883884266781880

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