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Properties

Label 2-418-209.189-c1-0-8
Degree $2$
Conductor $418$
Sign $0.996 - 0.0856i$
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 + (−0.238 − 0.328i)3-s + (−0.809 − 0.587i)4-s + (0.255 + 0.785i)5-s + (0.386 − 0.125i)6-s + (0.299 − 0.411i)7-s + (0.809 − 0.587i)8-s + (0.876 − 2.69i)9-s − 0.826·10-s + (1.68 − 2.85i)11-s + 0.406i·12-s + (0.131 − 0.405i)13-s + (0.299 + 0.411i)14-s + (0.197 − 0.271i)15-s + (0.309 + 0.951i)16-s + (1.95 − 0.634i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.137 − 0.189i)3-s + (−0.404 − 0.293i)4-s + (0.114 + 0.351i)5-s + (0.157 − 0.0512i)6-s + (0.113 − 0.155i)7-s + (0.286 − 0.207i)8-s + (0.292 − 0.898i)9-s − 0.261·10-s + (0.509 − 0.860i)11-s + 0.117i·12-s + (0.0365 − 0.112i)13-s + (0.0799 + 0.110i)14-s + (0.0509 − 0.0701i)15-s + (0.0772 + 0.237i)16-s + (0.473 − 0.153i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.996 - 0.0856i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.996 - 0.0856i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22228 + 0.0524633i\)
\(L(\frac12)\) \(\approx\) \(1.22228 + 0.0524633i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 2.85i)T \)
19 \( 1 + (1.29 - 4.16i)T \)
good3 \( 1 + (0.238 + 0.328i)T + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.255 - 0.785i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.299 + 0.411i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.131 + 0.405i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.95 + 0.634i)T + (13.7 - 9.99i)T^{2} \)
23 \( 1 - 2.95T + 23T^{2} \)
29 \( 1 + (0.182 + 0.132i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-8.75 - 2.84i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.56 + 7.65i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.56 + 1.86i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 6.90iT - 43T^{2} \)
47 \( 1 + (10.6 - 7.73i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.61 + 1.50i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.84 - 3.91i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.61 - 1.17i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 11.8iT - 67T^{2} \)
71 \( 1 + (6.28 - 2.04i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.67 - 5.05i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.46 - 10.6i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (7.08 - 2.30i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 14.9iT - 89T^{2} \)
97 \( 1 + (-2.41 - 0.785i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.12201575828151427567134533924, −10.21628960979538112872764559812, −9.298489353610167574222673557707, −8.421857348558527141903861292721, −7.42392109581491376506632713148, −6.44105195857510134217877783099, −5.87347326632330811739069684357, −4.41092742006734405393919907676, −3.19556889974030480481352342672, −1.04968646310350497669484166765, 1.46730069286626318579663938425, 2.82793983184073406026657960034, 4.45636590673396142061116498930, 4.98873222374521950670928591887, 6.53243191403478996075456869357, 7.67906039701976163871941117950, 8.615468271090055344799239996789, 9.585956230203967711161037553424, 10.22049748304002031660629357660, 11.25608316114532095570311911912

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