Properties

Label 2-418-19.9-c1-0-3
Degree $2$
Conductor $418$
Sign $-0.616 - 0.787i$
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  + (0.173 + 0.984i)2-s + (0.854 − 0.717i)3-s + (−0.939 + 0.342i)4-s + (−1.23 − 0.449i)5-s + (0.854 + 0.717i)6-s + (−2.13 + 3.70i)7-s + (−0.5 − 0.866i)8-s + (−0.304 + 1.72i)9-s + (0.227 − 1.29i)10-s + (0.5 + 0.866i)11-s + (−0.557 + 0.966i)12-s + (0.986 + 0.827i)13-s + (−4.01 − 1.46i)14-s + (−1.37 + 0.501i)15-s + (0.766 − 0.642i)16-s + (0.723 + 4.10i)17-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (0.493 − 0.414i)3-s + (−0.469 + 0.171i)4-s + (−0.551 − 0.200i)5-s + (0.349 + 0.292i)6-s + (−0.807 + 1.39i)7-s + (−0.176 − 0.306i)8-s + (−0.101 + 0.575i)9-s + (0.0720 − 0.408i)10-s + (0.150 + 0.261i)11-s + (−0.161 + 0.278i)12-s + (0.273 + 0.229i)13-s + (−1.07 − 0.390i)14-s + (−0.355 + 0.129i)15-s + (0.191 − 0.160i)16-s + (0.175 + 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.616 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.616 - 0.787i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.616 - 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.492728 + 1.01229i\)
\(L(\frac12)\) \(\approx\) \(0.492728 + 1.01229i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (4.32 - 0.545i)T \)
good3 \( 1 + (-0.854 + 0.717i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (1.23 + 0.449i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (2.13 - 3.70i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (-0.986 - 0.827i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.723 - 4.10i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-4.86 + 1.76i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.771 - 4.37i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (0.707 - 1.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.94T + 37T^{2} \)
41 \( 1 + (-1.09 + 0.919i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (7.34 + 2.67i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.24 + 7.06i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (6.61 - 2.40i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.314 - 1.78i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-5.07 + 1.84i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.669 + 3.79i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-4.55 - 1.65i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-8.09 + 6.78i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-5.53 + 4.64i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (5.90 - 10.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.59 + 1.34i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.63 + 9.28i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.76871065607407164425629653790, −10.57394537048435077910997883661, −9.301140702466397830128854098650, −8.582625828219102202030811994161, −8.016161056482066540131136173137, −6.80585900619408312205775000860, −6.00021037708405898568284368035, −4.87154357052639810334478934919, −3.53075903140029468260791009912, −2.20924289344803234149381147195, 0.66189199612107455702271442521, 2.96794137774737825790401086590, 3.69906342745920944400302911071, 4.48880677564078540229390144740, 6.18913439764443669234461272236, 7.21379676904942307254237963359, 8.247994843686182935832774000840, 9.468048184583303844112435621320, 9.842721806138236495075099091796, 11.00602579164428829876732999081

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