Properties

Label 2-418-11.4-c1-0-10
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: $\chi_{418} (191, \cdot )$
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

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

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