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Properties

Label 2-190-95.64-c1-0-5
Degree $2$
Conductor $190$
Sign $0.863 + 0.504i$
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  + (0.866 − 0.5i)2-s + (1.28 − 0.741i)3-s + (0.499 − 0.866i)4-s + (1.25 + 1.85i)5-s + (0.741 − 1.28i)6-s + 0.482i·7-s − 0.999i·8-s + (−0.400 + 0.693i)9-s + (2.01 + 0.977i)10-s − 4.43·11-s − 1.48i·12-s + (−3.58 − 2.07i)13-s + (0.241 + 0.418i)14-s + (2.98 + 1.44i)15-s + (−0.5 − 0.866i)16-s + (6.84 − 3.94i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.741 − 0.428i)3-s + (0.249 − 0.433i)4-s + (0.560 + 0.828i)5-s + (0.302 − 0.524i)6-s + 0.182i·7-s − 0.353i·8-s + (−0.133 + 0.231i)9-s + (0.635 + 0.309i)10-s − 1.33·11-s − 0.428i·12-s + (−0.994 − 0.574i)13-s + (0.0645 + 0.111i)14-s + (0.770 + 0.374i)15-s + (−0.125 − 0.216i)16-s + (1.65 − 0.957i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $0.863 + 0.504i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{190} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 190,\ (\ :1/2),\ 0.863 + 0.504i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89319 - 0.513127i\)
\(L(\frac12)\) \(\approx\) \(1.89319 - 0.513127i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-1.25 - 1.85i)T \)
19 \( 1 + (4.31 - 0.590i)T \)
good3 \( 1 + (-1.28 + 0.741i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 - 0.482iT - 7T^{2} \)
11 \( 1 + 4.43T + 11T^{2} \)
13 \( 1 + (3.58 + 2.07i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-6.84 + 3.94i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.90 + 2.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.91 + 3.31i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.58T + 31T^{2} \)
37 \( 1 - 3.50iT - 37T^{2} \)
41 \( 1 + (-4.44 - 7.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.551 - 0.318i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.69 + 2.13i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.76 + 1.59i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.812 - 1.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.735 - 1.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.4 - 6.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.41 + 2.44i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.95 - 3.43i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.576 + 0.997i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.4iT - 83T^{2} \)
89 \( 1 + (-4.37 + 7.58i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.825 + 0.476i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.69378134621840653785798718759, −11.65254227660241308881561292806, −10.28695855771330539030885575889, −9.968630365638964464041149391314, −8.169073850955227005452839776657, −7.45116009043892886425380751943, −6.03333101447212149220870080404, −4.97523878442873769570179354167, −2.93909131322998702685632055413, −2.41900186342184431470972375026, 2.41760999189937237835821760019, 3.91497937462934239642976648727, 5.08708575567655712240758926818, 6.10730394535217684883359565196, 7.73519039262432691101027277920, 8.476641722319977529751469908891, 9.663207317410521491785145467001, 10.41324631714165488383534706899, 12.15589087301757807695063507235, 12.65358898329012303256217053855

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