Properties

Label 2-6930-1.1-c1-0-8
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 − 3.46·13-s − 14-s + 16-s − 3.46·17-s + 20-s + 22-s − 5.46·23-s + 25-s + 3.46·26-s + 28-s − 2·29-s − 6.92·31-s − 32-s + 3.46·34-s + 35-s + 0.535·37-s − 40-s − 4.92·41-s + 10.9·43-s − 44-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 − 0.960·13-s − 0.267·14-s + 0.250·16-s − 0.840·17-s + 0.223·20-s + 0.213·22-s − 1.13·23-s + 0.200·25-s + 0.679·26-s + 0.188·28-s − 0.371·29-s − 1.24·31-s − 0.176·32-s + 0.594·34-s + 0.169·35-s + 0.0881·37-s − 0.158·40-s − 0.769·41-s + 1.66·43-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: 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

\(L(1)\) \(\approx\) \(1.164523443\)
\(L(\frac12)\) \(\approx\) \(1.164523443\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.46T + 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 5.46T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 6.92T + 31T^{2} \)
37 \( 1 - 0.535T + 37T^{2} \)
41 \( 1 + 4.92T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 - 0.928T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 - 8.92T + 61T^{2} \)
67 \( 1 - 2.53T + 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 - 12.9T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 6.92T + 83T^{2} \)
89 \( 1 - 3.46T + 89T^{2} \)
97 \( 1 - 15.8T + 97T^{2} \)
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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

−7.86817091426692951286676760735, −7.44159966918481628693241872822, −6.67645728759025185010571930130, −5.90624719315265733568864390077, −5.23563599171944562561386112902, −4.42277320494169648085506626055, −3.48121124671819618694571974185, −2.26015750238241324990905258330, −2.02277669445611681829690769554, −0.58675343213014481012918381257, 0.58675343213014481012918381257, 2.02277669445611681829690769554, 2.26015750238241324990905258330, 3.48121124671819618694571974185, 4.42277320494169648085506626055, 5.23563599171944562561386112902, 5.90624719315265733568864390077, 6.67645728759025185010571930130, 7.44159966918481628693241872822, 7.86817091426692951286676760735

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