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Properties

Label 2-418-11.9-c1-0-5
Degree $2$
Conductor $418$
Sign $0.187 - 0.982i$
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 + (1.27 − 0.929i)3-s + (−0.809 − 0.587i)4-s + (0.804 + 2.47i)5-s + (0.488 + 1.50i)6-s + (1.25 + 0.911i)7-s + (0.809 − 0.587i)8-s + (−0.154 + 0.474i)9-s − 2.60·10-s + (−1.72 + 2.83i)11-s − 1.58·12-s + (0.710 − 2.18i)13-s + (−1.25 + 0.911i)14-s + (3.32 + 2.41i)15-s + (0.309 + 0.951i)16-s + (0.0656 + 0.201i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.738 − 0.536i)3-s + (−0.404 − 0.293i)4-s + (0.359 + 1.10i)5-s + (0.199 + 0.614i)6-s + (0.474 + 0.344i)7-s + (0.286 − 0.207i)8-s + (−0.0513 + 0.158i)9-s − 0.822·10-s + (−0.521 + 0.853i)11-s − 0.456·12-s + (0.197 − 0.606i)13-s + (−0.335 + 0.243i)14-s + (0.859 + 0.624i)15-s + (0.0772 + 0.237i)16-s + (0.0159 + 0.0489i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.187 - 0.982i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.187 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22642 + 1.01444i\)
\(L(\frac12)\) \(\approx\) \(1.22642 + 1.01444i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 2.83i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (-1.27 + 0.929i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.804 - 2.47i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-1.25 - 0.911i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.710 + 2.18i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.0656 - 0.201i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 + (-3.79 - 2.75i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.22 + 3.75i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.14 - 1.55i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.10 + 4.43i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 5.85T + 43T^{2} \)
47 \( 1 + (10.1 - 7.33i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.31 + 7.12i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.38 + 2.45i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.02 + 12.3i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 8.97T + 67T^{2} \)
71 \( 1 + (-2.10 - 6.48i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-6.83 - 4.96i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.77 + 8.54i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (5.03 + 15.4i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 3.58T + 89T^{2} \)
97 \( 1 + (0.460 - 1.41i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.17677932707178040230245260647, −10.38072971540739805836135508050, −9.512974911656670713383617291846, −8.324573996397131642372962027886, −7.74412654432590953153921030284, −6.94208182374197689790109914876, −5.91917811943497739456430781517, −4.78224712306472749155342354492, −3.02215684868491107614034339325, −2.01228537470103780091853782759, 1.13838122697861426868247101162, 2.72893299167971553730786566939, 3.97927697435044935924276736166, 4.82577336382204729474912707040, 6.06971819630019873155627402841, 7.80997669615210734874949440227, 8.576331657370316228487521475394, 9.159360114487818307323453979436, 9.965772392091955623004898735624, 10.91340932757736578124014673635

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