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Properties

Label 2-190-95.64-c1-0-0
Degree $2$
Conductor $190$
Sign $-0.945 + 0.325i$
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 + (−0.590 + 0.341i)3-s + (0.499 − 0.866i)4-s + (−2.09 − 0.778i)5-s + (0.341 − 0.590i)6-s + 0.317i·7-s + 0.999i·8-s + (−1.26 + 2.19i)9-s + (2.20 − 0.373i)10-s − 4.31·11-s + 0.682i·12-s + (−5.45 − 3.14i)13-s + (−0.158 − 0.275i)14-s + (1.50 − 0.255i)15-s + (−0.5 − 0.866i)16-s + (0.115 − 0.0669i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.341 + 0.196i)3-s + (0.249 − 0.433i)4-s + (−0.937 − 0.348i)5-s + (0.139 − 0.241i)6-s + 0.120i·7-s + 0.353i·8-s + (−0.422 + 0.731i)9-s + (0.697 − 0.118i)10-s − 1.29·11-s + 0.196i·12-s + (−1.51 − 0.873i)13-s + (−0.0424 − 0.0735i)14-s + (0.388 − 0.0658i)15-s + (−0.125 − 0.216i)16-s + (0.0281 − 0.0162i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $-0.945 + 0.325i$
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: \((2,\ 190,\ (\ :1/2),\ -0.945 + 0.325i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00506031 - 0.0302129i\)
\(L(\frac12)\) \(\approx\) \(0.00506031 - 0.0302129i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.09 + 0.778i)T \)
19 \( 1 + (-4.05 + 1.60i)T \)
good3 \( 1 + (0.590 - 0.341i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 - 0.317iT - 7T^{2} \)
11 \( 1 + 4.31T + 11T^{2} \)
13 \( 1 + (5.45 + 3.14i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.115 + 0.0669i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.44 + 1.98i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.57 - 7.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.98T + 31T^{2} \)
37 \( 1 - 5.07iT - 37T^{2} \)
41 \( 1 + (-0.433 - 0.750i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.93 + 2.85i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-11.2 - 6.48i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.86 + 3.96i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.80 + 8.32i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.08 - 5.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.511 + 0.295i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.83 + 10.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.15 - 4.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.66 + 2.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.20iT - 83T^{2} \)
89 \( 1 + (1.85 - 3.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.20 - 2.42i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.87942647806867807612987897553, −12.03684575142770013957842722330, −10.92866245941894364692461795520, −10.26856143802430227948138087708, −9.006405968519456843445445684648, −7.77527509152090950828106231730, −7.48409255527704191880626725502, −5.51593170567030492625245810785, −4.87180049608963335143036135137, −2.77522177467690277633888468602, 0.03197929863616733925232349133, 2.60705396125501939874202551622, 4.06511954344431429929730972609, 5.69914336766947093195630649204, 7.29667328264709952235612730174, 7.68048934537643922726127171443, 9.123788005675366082216234475801, 10.08278919351224346609579528221, 11.15192482393813843896850641765, 11.93869692745962240830595562410

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