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Properties

Label 2-418-209.189-c1-0-2
Degree $2$
Conductor $418$
Sign $-0.855 - 0.517i$
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 + (−0.897 − 1.23i)3-s + (−0.809 − 0.587i)4-s + (1.27 + 3.91i)5-s + (1.45 − 0.471i)6-s + (−0.757 + 1.04i)7-s + (0.809 − 0.587i)8-s + (0.206 − 0.634i)9-s − 4.11·10-s + (−3.21 + 0.811i)11-s + 1.52i·12-s + (0.0655 − 0.201i)13-s + (−0.757 − 1.04i)14-s + (3.69 − 5.08i)15-s + (0.309 + 0.951i)16-s + (−3.89 + 1.26i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.518 − 0.713i)3-s + (−0.404 − 0.293i)4-s + (0.568 + 1.74i)5-s + (0.592 − 0.192i)6-s + (−0.286 + 0.393i)7-s + (0.286 − 0.207i)8-s + (0.0687 − 0.211i)9-s − 1.30·10-s + (−0.969 + 0.244i)11-s + 0.440i·12-s + (0.0181 − 0.0559i)13-s + (−0.202 − 0.278i)14-s + (0.953 − 1.31i)15-s + (0.0772 + 0.237i)16-s + (−0.943 + 0.306i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.855 - 0.517i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.855 - 0.517i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.188413 + 0.676292i\)
\(L(\frac12)\) \(\approx\) \(0.188413 + 0.676292i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.21 - 0.811i)T \)
19 \( 1 + (0.609 - 4.31i)T \)
good3 \( 1 + (0.897 + 1.23i)T + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (-1.27 - 3.91i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.757 - 1.04i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.0655 + 0.201i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (3.89 - 1.26i)T + (13.7 - 9.99i)T^{2} \)
23 \( 1 - 4.79T + 23T^{2} \)
29 \( 1 + (-5.64 - 4.09i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (9.68 + 3.14i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.68 - 5.07i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (9.07 - 6.59i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 2.10iT - 43T^{2} \)
47 \( 1 + (-6.59 + 4.78i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-10.6 - 3.46i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.23 + 8.58i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.79 + 1.23i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 4.28iT - 67T^{2} \)
71 \( 1 + (-0.965 + 0.313i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.84 + 2.54i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.22 - 6.84i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.32 + 1.07i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 7.99iT - 89T^{2} \)
97 \( 1 + (-5.26 - 1.70i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.42031930093474903989646046445, −10.55800276756618221281032872400, −9.943950956522876531565914296665, −8.752100786842171042399618297373, −7.45620866762271731798605211029, −6.84395760574958355515866931566, −6.21426886984230000658497743828, −5.34651836282671888869144740277, −3.42211829747345539903794040661, −2.07857974391129494780536985107, 0.49692797195129423691873028937, 2.23488666745230125526131731333, 4.06958152346635524458550269424, 4.98445783670094684321669609435, 5.46166061360506624297106207958, 7.20077770390821214912255331650, 8.622430590696099415657007449007, 9.035042897968426816419692927477, 10.07477495992954659508562969453, 10.67830504587403951492687929531

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