Properties

Label 2-190-19.9-c1-0-0
Degree $2$
Conductor $190$
Sign $-0.100 - 0.994i$
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 + (−0.849 + 0.712i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (0.849 + 0.712i)6-s + (−2.46 + 4.26i)7-s + (0.5 + 0.866i)8-s + (−0.307 + 1.74i)9-s + (−0.173 + 0.984i)10-s + (−2.20 − 3.82i)11-s + (0.554 − 0.960i)12-s + (2.02 + 1.69i)13-s + (4.63 + 1.68i)14-s + (1.04 − 0.379i)15-s + (0.766 − 0.642i)16-s + (0.872 + 4.94i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (−0.490 + 0.411i)3-s + (−0.469 + 0.171i)4-s + (−0.420 − 0.152i)5-s + (0.346 + 0.291i)6-s + (−0.931 + 1.61i)7-s + (0.176 + 0.306i)8-s + (−0.102 + 0.580i)9-s + (−0.0549 + 0.311i)10-s + (−0.665 − 1.15i)11-s + (0.160 − 0.277i)12-s + (0.560 + 0.470i)13-s + (1.23 + 0.450i)14-s + (0.269 − 0.0979i)15-s + (0.191 − 0.160i)16-s + (0.211 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $-0.100 - 0.994i$
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: \((2,\ 190,\ (\ :1/2),\ -0.100 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.326418 + 0.360922i\)
\(L(\frac12)\) \(\approx\) \(0.326418 + 0.360922i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.21 - 1.11i)T \)
good3 \( 1 + (0.849 - 0.712i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (2.46 - 4.26i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.20 + 3.82i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.02 - 1.69i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.872 - 4.94i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-0.964 + 0.351i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.462 + 2.62i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (1.01 - 1.76i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.00T + 37T^{2} \)
41 \( 1 + (-1.65 + 1.38i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (1.41 + 0.516i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.310 + 1.76i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (5.28 - 1.92i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-2.44 - 13.8i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-13.6 + 4.96i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (2.36 - 13.4i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-14.6 - 5.33i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (9.66 - 8.10i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (2.17 - 1.82i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-3.64 + 6.30i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.72 - 6.48i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-2.06 - 11.7i)T + (-91.1 + 33.1i)T^{2} \)
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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

−12.68824142368223590044321401133, −11.71225574909459604267872526594, −10.94923920143413684865242705205, −10.06516068858625773497006322277, −8.761910587232091112517899729629, −8.289899575094345758415180661800, −6.17333876671595660313017087067, −5.41995325713576458604314254350, −3.85986976887509476729195417761, −2.49133464522052084807702608092, 0.46809461467384583807723188460, 3.50493221301020625682751445743, 4.79605677690943472905919034905, 6.38177365161729971295560169287, 7.05410026856361874355059201798, 7.79322497724759122078642753453, 9.381727473306200360959904349692, 10.25908867295080593956432053326, 11.22003813347601607389158945076, 12.66033289715823612335071479238

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