Properties

Label 2-418-1.1-c1-0-9
Degree $2$
Conductor $418$
Sign $1$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
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 + 3·3-s + 4-s − 2·5-s + 3·6-s + 7-s + 8-s + 6·9-s − 2·10-s + 11-s + 3·12-s − 7·13-s + 14-s − 6·15-s + 16-s − 3·17-s + 6·18-s + 19-s − 2·20-s + 3·21-s + 22-s + 3·23-s + 3·24-s − 25-s − 7·26-s + 9·27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1/2·4-s − 0.894·5-s + 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s − 0.632·10-s + 0.301·11-s + 0.866·12-s − 1.94·13-s + 0.267·14-s − 1.54·15-s + 1/4·16-s − 0.727·17-s + 1.41·18-s + 0.229·19-s − 0.447·20-s + 0.654·21-s + 0.213·22-s + 0.625·23-s + 0.612·24-s − 1/5·25-s − 1.37·26-s + 1.73·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $1$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.054766158\)
\(L(\frac12)\) \(\approx\) \(3.054766158\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
11 \( 1 - T \)
19 \( 1 - T \)
good3 \( 1 - p T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−11.44031499387181880676869075421, −10.16945181356397758857550871516, −9.268473825100831991056204960490, −8.309421830301335966278787444926, −7.52862399314433966270679284406, −6.91725568718300574836338534653, −4.95214476220868061618204690309, −4.14835858272676134336675847541, −3.10644627478975717672239244823, −2.11774242997109846595062197067, 2.11774242997109846595062197067, 3.10644627478975717672239244823, 4.14835858272676134336675847541, 4.95214476220868061618204690309, 6.91725568718300574836338534653, 7.52862399314433966270679284406, 8.309421830301335966278787444926, 9.268473825100831991056204960490, 10.16945181356397758857550871516, 11.44031499387181880676869075421

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