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Properties

Label 2-418-11.5-c1-0-1
Degree $2$
Conductor $418$
Sign $0.870 - 0.491i$
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 + (−1.72 − 1.25i)3-s + (−0.809 + 0.587i)4-s + (−0.137 + 0.422i)5-s + (−0.657 + 2.02i)6-s + (−0.658 + 0.478i)7-s + (0.809 + 0.587i)8-s + (0.473 + 1.45i)9-s + 0.444·10-s + (−2.12 + 2.54i)11-s + 2.12·12-s + (0.164 + 0.505i)13-s + (0.658 + 0.478i)14-s + (0.765 − 0.556i)15-s + (0.309 − 0.951i)16-s + (−1.81 + 5.59i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.994 − 0.722i)3-s + (−0.404 + 0.293i)4-s + (−0.0614 + 0.189i)5-s + (−0.268 + 0.826i)6-s + (−0.248 + 0.180i)7-s + (0.286 + 0.207i)8-s + (0.157 + 0.485i)9-s + 0.140·10-s + (−0.642 + 0.766i)11-s + 0.614·12-s + (0.0455 + 0.140i)13-s + (0.175 + 0.127i)14-s + (0.197 − 0.143i)15-s + (0.0772 − 0.237i)16-s + (−0.440 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.870 - 0.491i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.870 - 0.491i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.508829 + 0.133643i\)
\(L(\frac12)\) \(\approx\) \(0.508829 + 0.133643i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.12 - 2.54i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
good3 \( 1 + (1.72 + 1.25i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.137 - 0.422i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (0.658 - 0.478i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.164 - 0.505i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.81 - 5.59i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 - 6.69T + 23T^{2} \)
29 \( 1 + (-3.37 + 2.44i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.79 + 5.51i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.92 - 1.39i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-8.29 - 6.02i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + (5.12 + 3.72i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.87 - 8.86i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.73 - 1.98i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.47 - 4.53i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 1.23T + 67T^{2} \)
71 \( 1 + (3.57 - 11.0i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (9.98 - 7.25i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.07 + 3.30i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.97 - 12.2i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 7.40T + 89T^{2} \)
97 \( 1 + (4.39 + 13.5i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.27218895403044091046163825624, −10.64216123090004082015439217374, −9.675293995561399946145588021961, −8.602356935740982992824291027453, −7.46625910638319542483911880786, −6.60747433150132320980024997313, −5.59559575849707672406485831724, −4.43152555183311462121518021072, −2.88471428168622723099146740956, −1.40834589597726837101860185962, 0.44247892754146385746450025311, 3.16604816267164516619883330140, 4.85963031047340277451867117228, 5.14558577562699563916812666901, 6.36847444640001018265035775901, 7.23697440023169340922568320438, 8.480524885388549866287878144864, 9.285907720707145865031625671661, 10.36792930405867098677327475804, 10.90395423751242427748366589318

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