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Properties

Label 2-418-11.5-c1-0-13
Degree $2$
Conductor $418$
Sign $-0.569 + 0.821i$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (−1 − 0.726i)3-s + (−0.809 + 0.587i)4-s + (−0.881 + 2.71i)5-s + (0.381 − 1.17i)6-s + (−2.30 + 1.67i)7-s + (−0.809 − 0.587i)8-s + (−0.454 − 1.40i)9-s − 2.85·10-s + (0.809 − 3.21i)11-s + 1.23·12-s + (−1.23 − 3.80i)13-s + (−2.30 − 1.67i)14-s + (2.85 − 2.07i)15-s + (0.309 − 0.951i)16-s + (−0.190 + 0.587i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.577 − 0.419i)3-s + (−0.404 + 0.293i)4-s + (−0.394 + 1.21i)5-s + (0.155 − 0.479i)6-s + (−0.872 + 0.634i)7-s + (−0.286 − 0.207i)8-s + (−0.151 − 0.466i)9-s − 0.902·10-s + (0.243 − 0.969i)11-s + 0.356·12-s + (−0.342 − 1.05i)13-s + (−0.617 − 0.448i)14-s + (0.736 − 0.535i)15-s + (0.0772 − 0.237i)16-s + (−0.0463 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.569 + 0.821i$
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: \(1\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.569 + 0.821i)\)

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 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-0.809 + 3.21i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (1 + 0.726i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.881 - 2.71i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.30 - 1.67i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.23 + 3.80i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.190 - 0.587i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 + 5.85T + 23T^{2} \)
29 \( 1 + (6.23 - 4.53i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.23 + 6.88i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1 - 0.726i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.85 - 4.97i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 5.38T + 43T^{2} \)
47 \( 1 + (4.73 + 3.44i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.09 - 6.43i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.85 + 6.43i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.42 - 7.46i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 4.76T + 67T^{2} \)
71 \( 1 + (2.23 - 6.88i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.61 - 4.08i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.47 + 4.53i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.73 + 8.42i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 8.18T + 89T^{2} \)
97 \( 1 + (0.291 + 0.898i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.08235241758570590935304329339, −9.998321643105668537789945792667, −8.966861998729850840285307015995, −7.83918011665768452616816068316, −6.94837103237608865023593866262, −6.09180975198479714640643138032, −5.67835710538149599116631558923, −3.72162461536278180979422466560, −2.91551217201481477856354777631, 0, 1.92941265174435522603516205395, 3.91355385149818967023482749797, 4.49366005419226709413236725954, 5.43652877721339042420783254296, 6.74252267014201928109645116424, 7.944136619377686248177857737582, 9.165771873766274980365724621792, 9.791617918060637591581250604234, 10.61117268631444804408550360229

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