Properties

Label 2-190-19.9-c1-0-4
Degree $2$
Conductor $190$
Sign $-0.436 + 0.899i$
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 + (−1.77 + 1.49i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (1.77 + 1.49i)6-s + (2.45 − 4.25i)7-s + (0.5 + 0.866i)8-s + (0.413 − 2.34i)9-s + (−0.173 + 0.984i)10-s + (−1.42 − 2.46i)11-s + (1.15 − 2.00i)12-s + (−5.02 − 4.21i)13-s + (−4.62 − 1.68i)14-s + (2.17 − 0.793i)15-s + (0.766 − 0.642i)16-s + (−0.380 − 2.15i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (−1.02 + 0.860i)3-s + (−0.469 + 0.171i)4-s + (−0.420 − 0.152i)5-s + (0.725 + 0.608i)6-s + (0.929 − 1.60i)7-s + (0.176 + 0.306i)8-s + (0.137 − 0.781i)9-s + (−0.0549 + 0.311i)10-s + (−0.429 − 0.743i)11-s + (0.334 − 0.579i)12-s + (−1.39 − 1.16i)13-s + (−1.23 − 0.449i)14-s + (0.562 − 0.204i)15-s + (0.191 − 0.160i)16-s + (−0.0922 − 0.523i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.436 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $-0.436 + 0.899i$
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: \((2,\ 190,\ (\ :1/2),\ -0.436 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.308735 - 0.493133i\)
\(L(\frac12)\) \(\approx\) \(0.308735 - 0.493133i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.17 - 1.26i)T \)
good3 \( 1 + (1.77 - 1.49i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-2.45 + 4.25i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.42 + 2.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.02 + 4.21i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.380 + 2.15i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-2.49 + 0.908i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.32 - 7.48i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.70 + 4.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.06T + 37T^{2} \)
41 \( 1 + (7.16 - 6.01i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (1.62 + 0.593i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.521 - 2.95i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-3.96 + 1.44i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.167 + 0.949i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-7.04 + 2.56i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.333 + 1.89i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-5.79 - 2.10i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (3.04 - 2.55i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-11.7 + 9.84i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-0.656 + 1.13i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.25 + 6.92i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.356 - 2.02i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.81828388060182057691796676992, −11.14600588952638556092741065339, −10.45322795004994383384618322648, −9.831010123893563279973255955181, −8.145319238601731026998857910464, −7.29812212225369261745422527407, −5.22126823264698630447258654665, −4.74487194476718124298461425556, −3.38702973498213077824883939747, −0.61985090747894710948866347094, 2.07792696166784797609453768899, 4.82316986218267432476115377757, 5.50072962936667785661716338491, 6.76657175028803888811751468521, 7.50875417387936890570407698952, 8.630162482099851455499452872450, 9.764790876212687062519761350257, 11.35096059213569471214520333912, 11.99508211847815180725143867851, 12.48717636828504814461380314579

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