↑

Properties

Label 2-190-95.64-c1-0-8
Degree $2$
Conductor $190$
Sign $0.496 + 0.868i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.590 − 0.341i)3-s + (0.499 − 0.866i)4-s + (0.373 − 2.20i)5-s + (0.341 − 0.590i)6-s − 0.317i·7-s − 0.999i·8-s + (−1.26 + 2.19i)9-s + (−0.778 − 2.09i)10-s − 4.31·11-s − 0.682i·12-s + (5.45 + 3.14i)13-s + (−0.158 − 0.275i)14-s + (−0.531 − 1.43i)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 + (0.167 − 0.985i)5-s + (0.139 − 0.241i)6-s − 0.120i·7-s − 0.353i·8-s + (−0.422 + 0.731i)9-s + (−0.246 − 0.662i)10-s − 1.29·11-s − 0.196i·12-s + (1.51 + 0.873i)13-s + (−0.0424 − 0.0735i)14-s + (−0.137 − 0.369i)15-s + (−0.125 − 0.216i)16-s + (−0.0281 + 0.0162i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $0.496 + 0.868i$
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: \((2,\ 190,\ (\ :1/2),\ 0.496 + 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51443 - 0.878389i\)
\(L(\frac12)\) \(\approx\) \(1.51443 - 0.878389i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.373 + 2.20i)T \)
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 \( 1 + (-4.20 + 2.42i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.68260147712905170246287438179, −11.39843152613334298024164389576, −10.72852400666721871663753833401, −9.304411667001255525804243902939, −8.455816125241365628399518396219, −7.29460673970616942297307257707, −5.66096136515767969282771973589, −4.91064085156419224268116974415, −3.36310949809395287536660938787, −1.72467669670311623532054966464, 2.81505695504687666332580165212, 3.64029449955467317108444418393, 5.48622887442447637733122623991, 6.26717837375177974926315586283, 7.57900272780624011493648348532, 8.481184704313251576430673831903, 9.863303366522314548530587733941, 10.85755016801611563797414565954, 11.70327140471290103158529045847, 13.08484214115239874111473286177

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