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Properties

Label 2-418-11.3-c1-0-0
Degree $2$
Conductor $418$
Sign $0.286 - 0.958i$
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 + (−0.897 − 2.76i)3-s + (0.309 − 0.951i)4-s + (2.41 + 1.75i)5-s + (2.34 + 1.70i)6-s + (−0.863 + 2.65i)7-s + (0.309 + 0.951i)8-s + (−4.39 + 3.19i)9-s − 2.98·10-s + (−0.874 + 3.19i)11-s − 2.90·12-s + (−4.94 + 3.59i)13-s + (−0.863 − 2.65i)14-s + (2.67 − 8.23i)15-s + (−0.809 − 0.587i)16-s + (4.10 + 2.98i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.518 − 1.59i)3-s + (0.154 − 0.475i)4-s + (1.07 + 0.783i)5-s + (0.959 + 0.696i)6-s + (−0.326 + 1.00i)7-s + (0.109 + 0.336i)8-s + (−1.46 + 1.06i)9-s − 0.942·10-s + (−0.263 + 0.964i)11-s − 0.838·12-s + (−1.37 + 0.996i)13-s + (−0.230 − 0.710i)14-s + (0.691 − 2.12i)15-s + (−0.202 − 0.146i)16-s + (0.995 + 0.723i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.286 - 0.958i$
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: \((2,\ 418,\ (\ :1/2),\ 0.286 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.595397 + 0.443410i\)
\(L(\frac12)\) \(\approx\) \(0.595397 + 0.443410i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.874 - 3.19i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (0.897 + 2.76i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.41 - 1.75i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.863 - 2.65i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (4.94 - 3.59i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.10 - 2.98i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + 4.21T + 23T^{2} \)
29 \( 1 + (1.43 - 4.41i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.38 + 3.18i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.61 - 8.04i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.22 + 6.84i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.90T + 43T^{2} \)
47 \( 1 + (0.120 + 0.371i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.57 + 5.50i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.48 + 10.7i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.39 - 3.19i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 9.66T + 67T^{2} \)
71 \( 1 + (6.39 + 4.64i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.42 - 7.44i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.21 - 0.880i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.57 + 3.32i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 + (9.75 - 7.08i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.63620654869422889201586822143, −10.19077729857779221969696301138, −9.724700411274707627860068466157, −8.473230714094746530753864453087, −7.36594948056477290418616360020, −6.78356562526554393211068823442, −6.04618941803097740643346640788, −5.24471826067389185698311210538, −2.41961133958152386369387075546, −1.88898624578359647840390987611, 0.58757067394522038224229013668, 2.88428482014318603355378957517, 4.08304579148322939135370266930, 5.22193017380753458330112956342, 5.84558664954288320092535840788, 7.52009326545739266246763885256, 8.672121766419874877438011069394, 9.757015469373503860847578549559, 9.976137696411317318449193063965, 10.56039454334673152218643103911

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