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Properties

Label 2-418-11.3-c1-0-11
Degree $2$
Conductor $418$
Sign $-0.215 + 0.976i$
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  + (0.809 − 0.587i)2-s + (−0.744 − 2.29i)3-s + (0.309 − 0.951i)4-s + (2.77 + 2.01i)5-s + (−1.94 − 1.41i)6-s + (0.326 − 1.00i)7-s + (−0.309 − 0.951i)8-s + (−2.26 + 1.64i)9-s + 3.43·10-s + (0.637 − 3.25i)11-s − 2.40·12-s + (0.266 − 0.193i)13-s + (−0.326 − 1.00i)14-s + (2.55 − 7.86i)15-s + (−0.809 − 0.587i)16-s + (2.82 + 2.04i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.429 − 1.32i)3-s + (0.154 − 0.475i)4-s + (1.24 + 0.902i)5-s + (−0.795 − 0.577i)6-s + (0.123 − 0.379i)7-s + (−0.109 − 0.336i)8-s + (−0.755 + 0.548i)9-s + 1.08·10-s + (0.192 − 0.981i)11-s − 0.695·12-s + (0.0738 − 0.0536i)13-s + (−0.0872 − 0.268i)14-s + (0.659 − 2.03i)15-s + (−0.202 − 0.146i)16-s + (0.684 + 0.496i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.215 + 0.976i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.215 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24470 - 1.54931i\)
\(L(\frac12)\) \(\approx\) \(1.24470 - 1.54931i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.637 + 3.25i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (0.744 + 2.29i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.77 - 2.01i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.326 + 1.00i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.266 + 0.193i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.82 - 2.04i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + 7.90T + 23T^{2} \)
29 \( 1 + (1.91 - 5.89i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.78 - 1.29i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.98 + 6.10i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.26 + 3.88i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.47T + 43T^{2} \)
47 \( 1 + (-3.18 - 9.80i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.313 - 0.227i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.235 - 0.725i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.77 - 2.01i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 + (-10.1 - 7.40i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.34 - 4.14i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-12.2 + 8.90i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.76 - 5.64i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 6.50T + 89T^{2} \)
97 \( 1 + (-3.88 + 2.82i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.92742959263146034868690773560, −10.45989042420471464392432307055, −9.326092901421108940376833914515, −7.86918323889028852563716633881, −6.94395444864420684879488788856, −5.96407532094455546174503607862, −5.73784065853850524738322800127, −3.72077050195074842850593254621, −2.38587261832254410321862113557, −1.31637325951306153208624672463, 2.12527627548866563137543671478, 3.94172306940222660437012476315, 4.85388516341643636967688920305, 5.47683539137570688238402604212, 6.27841415375831148667659333120, 7.80308483755859113701003980007, 9.045021474412870504441091298421, 9.718924133935561965500019798598, 10.24726517690954173231490956730, 11.63675231517685402847186081739

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