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Properties

Label 2-190-95.64-c1-0-2
Degree $2$
Conductor $190$
Sign $0.582 - 0.813i$
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 + (1.15 − 0.664i)3-s + (0.499 − 0.866i)4-s + (−0.866 + 2.06i)5-s + (−0.664 + 1.15i)6-s + 2.32i·7-s + 0.999i·8-s + (−0.616 + 1.06i)9-s + (−0.280 − 2.21i)10-s + 6.39·11-s − 1.32i·12-s + (−0.743 − 0.429i)13-s + (−1.16 − 2.01i)14-s + (0.372 + 2.94i)15-s + (−0.5 − 0.866i)16-s + (4.06 − 2.34i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.664 − 0.383i)3-s + (0.249 − 0.433i)4-s + (−0.387 + 0.921i)5-s + (−0.271 + 0.469i)6-s + 0.880i·7-s + 0.353i·8-s + (−0.205 + 0.355i)9-s + (−0.0885 − 0.701i)10-s + 1.92·11-s − 0.383i·12-s + (−0.206 − 0.119i)13-s + (−0.311 − 0.539i)14-s + (0.0961 + 0.761i)15-s + (−0.125 − 0.216i)16-s + (0.985 − 0.568i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $0.582 - 0.813i$
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: \((2,\ 190,\ (\ :1/2),\ 0.582 - 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.943035 + 0.484644i\)
\(L(\frac12)\) \(\approx\) \(0.943035 + 0.484644i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 2.06i)T \)
19 \( 1 + (3.75 + 2.21i)T \)
good3 \( 1 + (-1.15 + 0.664i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 - 2.32iT - 7T^{2} \)
11 \( 1 - 6.39T + 11T^{2} \)
13 \( 1 + (0.743 + 0.429i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.06 + 2.34i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.00 - 1.73i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.21 - 3.82i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.25T + 31T^{2} \)
37 \( 1 + 9.76iT - 37T^{2} \)
41 \( 1 + (1.84 + 3.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.17 + 3.56i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.22 + 3.59i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.37 - 3.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.09 + 5.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.01 + 6.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.24 + 2.45i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.10 + 1.90i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.03 - 2.32i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.79 - 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.07iT - 83T^{2} \)
89 \( 1 + (-5.64 + 9.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.82 + 5.67i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.59378972577762920366165592301, −11.58600938366420588377826403820, −10.79042244310561551403608166227, −9.302174366376466262494723952428, −8.808627574026699339928559636142, −7.53283509597777577238575721246, −6.85285543302774841045920412591, −5.56954314442606368492035622230, −3.52336527234129970152584276345, −2.10915673935487434084590221172, 1.28062591285672985550731613511, 3.59229237505052533925202032508, 4.25205689581004170771039916553, 6.30349337699509095053830826589, 7.63531443510040157329986804055, 8.663616723984648034351201224583, 9.284713087753935406401406724559, 10.18542217300364369788650318465, 11.50874950062338506439812029242, 12.20517896651057814138350473327

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