Properties

Label 2-418-1.1-c1-0-4
Degree $2$
Conductor $418$
Sign $1$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2.56·3-s + 4-s + 2·5-s − 2.56·6-s − 0.561·7-s − 8-s + 3.56·9-s − 2·10-s + 11-s + 2.56·12-s + 0.561·13-s + 0.561·14-s + 5.12·15-s + 16-s − 0.561·17-s − 3.56·18-s − 19-s + 2·20-s − 1.43·21-s − 22-s + 1.43·23-s − 2.56·24-s − 25-s − 0.561·26-s + 1.43·27-s − 0.561·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.47·3-s + 0.5·4-s + 0.894·5-s − 1.04·6-s − 0.212·7-s − 0.353·8-s + 1.18·9-s − 0.632·10-s + 0.301·11-s + 0.739·12-s + 0.155·13-s + 0.150·14-s + 1.32·15-s + 0.250·16-s − 0.136·17-s − 0.839·18-s − 0.229·19-s + 0.447·20-s − 0.313·21-s − 0.213·22-s + 0.299·23-s − 0.522·24-s − 0.200·25-s − 0.110·26-s + 0.276·27-s − 0.106·28-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: no
Arithmetic: yes
Character: $\chi_{418} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.808717920\)
\(L(\frac12)\) \(\approx\) \(1.808717920\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.56T + 3T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
7 \( 1 + 0.561T + 7T^{2} \)
13 \( 1 - 0.561T + 13T^{2} \)
17 \( 1 + 0.561T + 17T^{2} \)
23 \( 1 - 1.43T + 23T^{2} \)
29 \( 1 + 5.68T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 5.12T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 12.8T + 53T^{2} \)
59 \( 1 + 7.68T + 59T^{2} \)
61 \( 1 - 6.24T + 61T^{2} \)
67 \( 1 + 7.68T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 9.68T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 + 0.876T + 89T^{2} \)
97 \( 1 + 7.12T + 97T^{2} \)
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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

−10.86878081460772726630137738357, −9.938850994382666261086169202310, −9.269548033660763706910973034590, −8.700309598822903808568055572210, −7.74131100109873532120571409940, −6.79288055299980896065481533675, −5.65341339453139235578459003044, −3.91222257259820018576286398452, −2.72121495121319900109329876429, −1.72927659535003427204067869921, 1.72927659535003427204067869921, 2.72121495121319900109329876429, 3.91222257259820018576286398452, 5.65341339453139235578459003044, 6.79288055299980896065481533675, 7.74131100109873532120571409940, 8.700309598822903808568055572210, 9.269548033660763706910973034590, 9.938850994382666261086169202310, 10.86878081460772726630137738357

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