Properties

Label 2-6930-1.1-c1-0-17
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 − 2·19-s + 20-s + 22-s + 4·23-s + 25-s − 2·26-s − 28-s + 10·29-s − 4·31-s − 32-s − 35-s − 4·37-s + 2·38-s − 40-s − 6·41-s + 6·43-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 − 0.458·19-s + 0.223·20-s + 0.213·22-s + 0.834·23-s + 1/5·25-s − 0.392·26-s − 0.188·28-s + 1.85·29-s − 0.718·31-s − 0.176·32-s − 0.169·35-s − 0.657·37-s + 0.324·38-s − 0.158·40-s − 0.937·41-s + 0.914·43-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

\(L(1)\) \(\approx\) \(1.419032292\)
\(L(\frac12)\) \(\approx\) \(1.419032292\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.263783043118445658627637948939, −7.06277172352689287665702293193, −6.80497317566770523206681840269, −5.94177224906595788916007559231, −5.29957965689161254804884714096, −4.36707879462100514588224343088, −3.34711559818770167127130244291, −2.64536652094341162060297480995, −1.69961732288693581850333296609, −0.68508989103098903077156499148, 0.68508989103098903077156499148, 1.69961732288693581850333296609, 2.64536652094341162060297480995, 3.34711559818770167127130244291, 4.36707879462100514588224343088, 5.29957965689161254804884714096, 5.94177224906595788916007559231, 6.80497317566770523206681840269, 7.06277172352689287665702293193, 8.263783043118445658627637948939

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