Properties

Label 2-6930-1.1-c1-0-92
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 + 3.75·13-s + 14-s + 16-s + 1.75·17-s − 8.12·19-s − 20-s − 22-s − 7.14·23-s + 25-s + 3.75·26-s + 28-s − 9.51·29-s + 6.12·31-s + 32-s + 1.75·34-s − 35-s − 5.75·37-s − 8.12·38-s − 40-s + 4.12·41-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 + 1.04·13-s + 0.267·14-s + 0.250·16-s + 0.426·17-s − 1.86·19-s − 0.223·20-s − 0.213·22-s − 1.49·23-s + 0.200·25-s + 0.737·26-s + 0.188·28-s − 1.76·29-s + 1.10·31-s + 0.176·32-s + 0.301·34-s − 0.169·35-s − 0.946·37-s − 1.31·38-s − 0.158·40-s + 0.644·41-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 \( 1 - 3.75T + 13T^{2} \)
17 \( 1 - 1.75T + 17T^{2} \)
19 \( 1 + 8.12T + 19T^{2} \)
23 \( 1 + 7.14T + 23T^{2} \)
29 \( 1 + 9.51T + 29T^{2} \)
31 \( 1 - 6.12T + 31T^{2} \)
37 \( 1 + 5.75T + 37T^{2} \)
41 \( 1 - 4.12T + 41T^{2} \)
43 \( 1 + 8.12T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 0.610T + 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 - 4.98T + 67T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 + 8.12T + 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 - 8.90T + 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
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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

−7.76104153668139530038318015318, −6.66490940519003090322982407158, −6.22315025920556225544951061987, −5.46660221404255909144738983369, −4.65177129203662840285166443731, −3.94816402854036180245483126776, −3.44261196050074013296157654432, −2.28445690334232451786152707223, −1.55207668086363353708547768096, 0, 1.55207668086363353708547768096, 2.28445690334232451786152707223, 3.44261196050074013296157654432, 3.94816402854036180245483126776, 4.65177129203662840285166443731, 5.46660221404255909144738983369, 6.22315025920556225544951061987, 6.66490940519003090322982407158, 7.76104153668139530038318015318

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