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Properties

Label 2-418-209.189-c1-0-6
Degree $2$
Conductor $418$
Sign $-0.929 - 0.369i$
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 + (1.52 + 2.10i)3-s + (−0.809 − 0.587i)4-s + (0.852 + 2.62i)5-s + (−2.47 + 0.803i)6-s + (−1.95 + 2.68i)7-s + (0.809 − 0.587i)8-s + (−1.16 + 3.57i)9-s − 2.76·10-s + (3.28 − 0.478i)11-s − 2.59i·12-s + (2.20 − 6.77i)13-s + (−1.95 − 2.68i)14-s + (−4.21 + 5.80i)15-s + (0.309 + 0.951i)16-s + (−4.75 + 1.54i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.882 + 1.21i)3-s + (−0.404 − 0.293i)4-s + (0.381 + 1.17i)5-s + (−1.00 + 0.327i)6-s + (−0.737 + 1.01i)7-s + (0.286 − 0.207i)8-s + (−0.387 + 1.19i)9-s − 0.872·10-s + (0.989 − 0.144i)11-s − 0.750i·12-s + (0.610 − 1.87i)13-s + (−0.521 − 0.718i)14-s + (−1.08 + 1.49i)15-s + (0.0772 + 0.237i)16-s + (−1.15 + 0.374i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.929 - 0.369i$
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: \((2,\ 418,\ (\ :1/2),\ -0.929 - 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.301088 + 1.57146i\)
\(L(\frac12)\) \(\approx\) \(0.301088 + 1.57146i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.28 + 0.478i)T \)
19 \( 1 + (0.551 + 4.32i)T \)
good3 \( 1 + (-1.52 - 2.10i)T + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.852 - 2.62i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (1.95 - 2.68i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-2.20 + 6.77i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.75 - 1.54i)T + (13.7 - 9.99i)T^{2} \)
23 \( 1 - 6.23T + 23T^{2} \)
29 \( 1 + (-2.53 - 1.84i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.98 + 0.646i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.78 - 2.45i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.76 + 1.28i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.76iT - 43T^{2} \)
47 \( 1 + (5.86 - 4.26i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (9.05 + 2.94i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.181 - 0.249i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.06 - 1.64i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 3.89iT - 67T^{2} \)
71 \( 1 + (-9.57 + 3.11i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.63 - 3.62i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.0577 + 0.177i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-14.3 + 4.67i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 7.75iT - 89T^{2} \)
97 \( 1 + (-8.91 - 2.89i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.03953297592789156417620300497, −10.56727112663239307120522407117, −9.503299286842973375722767420338, −9.033730312025384214845340003083, −8.228021594761017455461159484174, −6.75436477612198824056445997155, −6.10435719563530542216999203861, −4.86356339774228707863098658422, −3.37412902124038151262601361276, −2.80014890891512921670649584065, 1.13758766241572507032569498105, 1.95009355041864619232166457293, 3.61865126517399553920379944440, 4.55115374763001241514645921378, 6.52237524245735482579386837134, 7.01611021522591188491578571112, 8.323906688697621587792574424063, 9.143286255901159807929574511351, 9.415624144873041925621124128164, 10.91111772502052373688944965857

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