Properties

Label 2-190-19.9-c1-0-7
Degree $2$
Conductor $190$
Sign $-0.287 + 0.957i$
Analytic cond. $1.51715$
Root an. cond. $1.23172$
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 + (2.62 − 2.20i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (−2.62 − 2.20i)6-s + (−0.933 + 1.61i)7-s + (0.5 + 0.866i)8-s + (1.52 − 8.62i)9-s + (−0.173 + 0.984i)10-s + (1.80 + 3.12i)11-s + (−1.71 + 2.96i)12-s + (−2.17 − 1.82i)13-s + (1.75 + 0.638i)14-s + (−3.22 + 1.17i)15-s + (0.766 − 0.642i)16-s + (1.03 + 5.89i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (1.51 − 1.27i)3-s + (−0.469 + 0.171i)4-s + (−0.420 − 0.152i)5-s + (−1.07 − 0.899i)6-s + (−0.352 + 0.611i)7-s + (0.176 + 0.306i)8-s + (0.506 − 2.87i)9-s + (−0.0549 + 0.311i)10-s + (0.543 + 0.941i)11-s + (−0.494 + 0.857i)12-s + (−0.602 − 0.505i)13-s + (0.468 + 0.170i)14-s + (−0.831 + 0.302i)15-s + (0.191 − 0.160i)16-s + (0.252 + 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $-0.287 + 0.957i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{190} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 190,\ (\ :1/2),\ -0.287 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.908608 - 1.22138i\)
\(L(\frac12)\) \(\approx\) \(0.908608 - 1.22138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-4.32 - 0.577i)T \)
good3 \( 1 + (-2.62 + 2.20i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (0.933 - 1.61i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.80 - 3.12i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.17 + 1.82i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.03 - 5.89i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (4.78 - 1.74i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.204 + 1.16i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.59 + 4.50i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.35T + 37T^{2} \)
41 \( 1 + (2.85 - 2.39i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.631 + 0.229i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.09 - 6.22i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (2.40 - 0.876i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.827 + 4.69i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (7.73 - 2.81i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.38 + 7.84i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-1.81 - 0.659i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (3.83 - 3.22i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (0.460 - 0.386i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-0.951 + 1.64i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.755 - 0.633i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (0.0325 + 0.184i)T + (-91.1 + 33.1i)T^{2} \)
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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

−12.38232115242453593588879387100, −11.84545751649749131616602240421, −9.885598881067074130674532781961, −9.279146086342590270293062219828, −8.128071874790617996840502051779, −7.59667118836382251852359436341, −6.20914334684944471180110348398, −3.99780242636599874111702404593, −2.85413822781681200940261663491, −1.62181826236487701499298767948, 3.02385895421583769238087299505, 4.00618113449670693275826660401, 5.08647250200798747219867920243, 7.02100376132560323480217224778, 7.910892374970017100472733874747, 8.893735980326067305166777728338, 9.634394459409895024382380129072, 10.42089448325651155430366809415, 11.72379995591505961209038413411, 13.57526078480269663916731990257

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