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Properties

Label 2-418-19.11-c1-0-13
Degree $2$
Conductor $418$
Sign $-0.875 + 0.483i$
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  + (0.5 − 0.866i)2-s + (1.22 − 2.12i)3-s + (−0.499 − 0.866i)4-s + (−1.22 − 2.12i)6-s − 2.44·7-s − 0.999·8-s + (−1.49 − 2.59i)9-s − 11-s − 2.44·12-s + (−1.94 − 3.37i)13-s + (−1.22 + 2.12i)14-s + (−0.5 + 0.866i)16-s + (2.22 − 3.85i)17-s − 2.99·18-s + (4.17 − 1.25i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.707 − 1.22i)3-s + (−0.249 − 0.433i)4-s + (−0.499 − 0.866i)6-s − 0.925·7-s − 0.353·8-s + (−0.499 − 0.866i)9-s − 0.301·11-s − 0.707·12-s + (−0.540 − 0.936i)13-s + (−0.327 + 0.566i)14-s + (−0.125 + 0.216i)16-s + (0.539 − 0.934i)17-s − 0.707·18-s + (0.957 − 0.287i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 + 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.875 + 0.483i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.875 + 0.483i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.428498 - 1.66089i\)
\(L(\frac12)\) \(\approx\) \(0.428498 - 1.66089i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + T \)
19 \( 1 + (-4.17 + 1.25i)T \)
good3 \( 1 + (-1.22 + 2.12i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 2.44T + 7T^{2} \)
13 \( 1 + (1.94 + 3.37i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.22 + 3.85i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.449 + 0.778i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.949 - 1.64i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 8.44T + 37T^{2} \)
41 \( 1 + (-3.77 + 6.53i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.72 - 8.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.27 - 5.67i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.89 - 6.75i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.44 - 2.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.94 + 6.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.77 + 8.27i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.72 - 6.45i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.22 + 12.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.22 - 3.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.44T + 83T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.94 - 3.37i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.93603085245967551173163314327, −9.824504895675188457082977850874, −9.163788450714720811237210899195, −7.85540533194878010393765174010, −7.25660365171826809343233341761, −6.12044375209762311012044547932, −4.96917511571914993564749073160, −3.16443933309406537574191905229, −2.68581937347491886501962845288, −0.957804642011106021142396008386, 2.78419605402667272836262289134, 3.77517802349707257875465153411, 4.61365356222814938401413189332, 5.80484191065818394077837693307, 6.86415560382151098919822035226, 8.023850597973445075790888616251, 8.911305927003882938186547876910, 9.811305908631027603734514414341, 10.20001349336507628709386955925, 11.64820403706950868328213684517

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