Properties

Label 2-6930-1.1-c1-0-15
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 − 6·13-s − 14-s + 16-s + 6·19-s − 20-s − 22-s + 4·23-s + 25-s − 6·26-s − 28-s − 2·29-s + 4·31-s + 32-s + 35-s − 12·37-s + 6·38-s − 40-s − 2·41-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 − 1.66·13-s − 0.267·14-s + 1/4·16-s + 1.37·19-s − 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s − 1.17·26-s − 0.188·28-s − 0.371·29-s + 0.718·31-s + 0.176·32-s + 0.169·35-s − 1.97·37-s + 0.973·38-s − 0.158·40-s − 0.312·41-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

\(L(1)\) \(\approx\) \(2.371786783\)
\(L(\frac12)\) \(\approx\) \(2.371786783\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−7.68801916614768064875116732365, −7.17024019376813505573949730864, −6.72834658068974033615884271922, −5.55519902248088353357844803098, −5.16298382393628690453196022531, −4.45533180679141005357226466401, −3.49612460740407537991827297278, −2.92021179936521900202861423853, −2.07673172439245221003490464105, −0.68369410251442235037880369878, 0.68369410251442235037880369878, 2.07673172439245221003490464105, 2.92021179936521900202861423853, 3.49612460740407537991827297278, 4.45533180679141005357226466401, 5.16298382393628690453196022531, 5.55519902248088353357844803098, 6.72834658068974033615884271922, 7.17024019376813505573949730864, 7.68801916614768064875116732365

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