Properties

Label 2-22386-1.1-c1-0-3
Degree $2$
Conductor $22386$
Sign $1$
Analytic cond. $178.753$
Root an. cond. $13.3698$
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 − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 6·11-s − 12-s + 13-s + 14-s − 15-s + 16-s + 4·17-s − 18-s − 4·19-s + 20-s + 21-s − 6·22-s − 3·23-s + 24-s − 4·25-s − 26-s − 27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.80·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s − 1.27·22-s − 0.625·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22386\)    =    \(2 \cdot 3 \cdot 7 \cdot 13 \cdot 41\)
Sign: $1$
Analytic conductor: \(178.753\)
Root analytic conductor: \(13.3698\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 22386,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.218110055\)
\(L(\frac12)\) \(\approx\) \(1.218110055\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 - T \)
41 \( 1 + T \)
good5 \( 1 - T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.65786571954006, −14.99451585421339, −14.47223023406601, −14.01034185231725, −13.27312398400065, −12.66032920210978, −12.11285238305372, −11.63622606843137, −11.18981849105301, −10.43802825980040, −9.945203812584494, −9.504349159778867, −8.943391071656447, −8.395235432318970, −7.562238626075246, −7.028686860555599, −6.336099011771414, −6.010344389623342, −5.456660690529011, −4.348852843962109, −3.830422210781875, −3.127328784010228, −1.885539513055203, −1.563455225815947, −0.5274573708810066, 0.5274573708810066, 1.563455225815947, 1.885539513055203, 3.127328784010228, 3.830422210781875, 4.348852843962109, 5.456660690529011, 6.010344389623342, 6.336099011771414, 7.028686860555599, 7.562238626075246, 8.395235432318970, 8.943391071656447, 9.504349159778867, 9.945203812584494, 10.43802825980040, 11.18981849105301, 11.63622606843137, 12.11285238305372, 12.66032920210978, 13.27312398400065, 14.01034185231725, 14.47223023406601, 14.99451585421339, 15.65786571954006

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