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Properties

Label 2-418-11.5-c1-0-17
Degree $2$
Conductor $418$
Sign $0.719 + 0.694i$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.881 − 2.71i)5-s + (1.30 − 0.951i)7-s + (−0.809 − 0.587i)8-s + (−0.927 − 2.85i)9-s + 2.85·10-s + (−3.04 − 1.31i)11-s + (−2 − 6.15i)13-s + (1.30 + 0.951i)14-s + (0.309 − 0.951i)16-s + (−2.19 + 6.74i)17-s + (2.42 − 1.76i)18-s + (−0.809 − 0.587i)19-s + (0.881 + 2.71i)20-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (0.394 − 1.21i)5-s + (0.494 − 0.359i)7-s + (−0.286 − 0.207i)8-s + (−0.309 − 0.951i)9-s + 0.902·10-s + (−0.918 − 0.396i)11-s + (−0.554 − 1.70i)13-s + (0.349 + 0.254i)14-s + (0.0772 − 0.237i)16-s + (−0.531 + 1.63i)17-s + (0.572 − 0.415i)18-s + (−0.185 − 0.134i)19-s + (0.197 + 0.606i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.719 + 0.694i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.719 + 0.694i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29686 - 0.523545i\)
\(L(\frac12)\) \(\approx\) \(1.29686 - 0.523545i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (3.04 + 1.31i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.881 + 2.71i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.30 + 0.951i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (2 + 6.15i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.19 - 6.74i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 - 7.85T + 23T^{2} \)
29 \( 1 + (-2.61 + 1.90i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.38 - 7.33i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.85 + 2.80i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.23 + 0.898i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 0.618T + 43T^{2} \)
47 \( 1 + (-7.59 - 5.51i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.618 - 1.90i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.09 + 5.87i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.336 - 1.03i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + (0.145 - 0.449i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.381 - 0.277i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.70 + 11.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.881 - 2.71i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 8.18T + 89T^{2} \)
97 \( 1 + (5.09 + 15.6i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.97438948930478800963000684739, −10.16526284074638926470062066308, −8.883868377391657285815345371748, −8.436712610083848133007109235980, −7.49627530444988714730757975594, −6.13108460847911043587454809768, −5.35481768084013896823237778449, −4.52951266680804488479275254185, −3.05709982234276060185523615154, −0.860856831281497763554662376782, 2.28573191949558345693742648555, 2.66694860294496584681334085296, 4.56037786507942297073302732816, 5.26546451173855260370565130317, 6.71881898000411231284983709597, 7.46220679469244254824410716168, 8.804307239983112982996551053528, 9.723369325123647609238436270736, 10.58069397604981323884425522376, 11.35126731987160974985805176734

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