Properties

Label 2-22386-1.1-c1-0-26
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 $2$

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 + 11-s − 12-s + 13-s + 14-s + 15-s + 16-s − 5·17-s − 18-s − 2·19-s − 20-s + 21-s − 22-s + 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 + 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.218·21-s − 0.213·22-s + 0.208·23-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: \(2\)
Selberg data: \((2,\ 22386,\ (\ :1/2),\ 1)\)

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 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 - T \)
41 \( 1 + T \)
good5 \( 1 + T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−16.12062146293561, −15.60820937119894, −15.15619069627405, −14.67726780904120, −13.83445816865554, −13.10373746067717, −12.85647304319360, −12.10758615109728, −11.51396597980169, −11.05266847853735, −10.80165291106484, −9.868349653014357, −9.500398196353226, −8.857662769044121, −8.316855809696195, −7.651444184838129, −6.840763815662750, −6.767137626739585, −5.823567112002288, −5.357416412252870, −4.392329562365335, −3.805575619150327, −3.154853682130229, −1.997220347789467, −1.545854151712534, 0, 0, 1.545854151712534, 1.997220347789467, 3.154853682130229, 3.805575619150327, 4.392329562365335, 5.357416412252870, 5.823567112002288, 6.767137626739585, 6.840763815662750, 7.651444184838129, 8.316855809696195, 8.857662769044121, 9.500398196353226, 9.868349653014357, 10.80165291106484, 11.05266847853735, 11.51396597980169, 12.10758615109728, 12.85647304319360, 13.10373746067717, 13.83445816865554, 14.67726780904120, 15.15619069627405, 15.60820937119894, 16.12062146293561

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