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Properties

Label 2-418-11.3-c1-0-6
Degree $2$
Conductor $418$
Sign $0.877 - 0.479i$
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 + (0.540 + 1.66i)3-s + (0.309 − 0.951i)4-s + (2.42 + 1.76i)5-s + (1.41 + 1.02i)6-s + (0.0678 − 0.208i)7-s + (−0.309 − 0.951i)8-s + (−0.0511 + 0.0371i)9-s + 2.99·10-s + (−2.87 + 1.65i)11-s + 1.75·12-s + (1.17 − 0.855i)13-s + (−0.0678 − 0.208i)14-s + (−1.62 + 4.98i)15-s + (−0.809 − 0.587i)16-s + (−2.45 − 1.78i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.312 + 0.961i)3-s + (0.154 − 0.475i)4-s + (1.08 + 0.787i)5-s + (0.578 + 0.419i)6-s + (0.0256 − 0.0788i)7-s + (−0.109 − 0.336i)8-s + (−0.0170 + 0.0123i)9-s + 0.947·10-s + (−0.865 + 0.500i)11-s + 0.505·12-s + (0.326 − 0.237i)13-s + (−0.0181 − 0.0557i)14-s + (−0.418 + 1.28i)15-s + (−0.202 − 0.146i)16-s + (−0.594 − 0.432i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.877 - 0.479i$
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: \((2,\ 418,\ (\ :1/2),\ 0.877 - 0.479i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28758 + 0.584421i\)
\(L(\frac12)\) \(\approx\) \(2.28758 + 0.584421i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.87 - 1.65i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (-0.540 - 1.66i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.42 - 1.76i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.0678 + 0.208i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.17 + 0.855i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.45 + 1.78i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + 1.59T + 23T^{2} \)
29 \( 1 + (-1.09 + 3.36i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.58 - 1.87i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.324 + 0.997i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.77 + 5.45i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.67T + 43T^{2} \)
47 \( 1 + (2.13 + 6.58i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.20 - 0.879i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.0621 - 0.191i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (9.14 + 6.64i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 5.87T + 67T^{2} \)
71 \( 1 + (2.77 + 2.01i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.83 + 8.72i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.19 - 3.05i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.18 + 3.03i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 6.32T + 89T^{2} \)
97 \( 1 + (3.41 - 2.48i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.90041661934014036375149571551, −10.40412650546447831141769047557, −9.790449318458357418911485313559, −8.947149169427389704495469685161, −7.42357325016324663685438052680, −6.32576504535713407599941112324, −5.34857462914238744231713900481, −4.32175709704897493594157106819, −3.15116315620410360200536654843, −2.13281082225622641374778857359, 1.56704064725194415016527443158, 2.71530751464004419306391298248, 4.46640135650679432026264410656, 5.54679481443352215466076981358, 6.29044977567287602692646431420, 7.34997955097397604033248203306, 8.301060425732259939339496007022, 9.008101403662238846948807292962, 10.21276145071504631062227773751, 11.29730419474383873539690889898

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