Properties

Label 2-190-19.16-c1-0-2
Degree $2$
Conductor $190$
Sign $0.997 - 0.0671i$
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  + (−0.766 − 0.642i)2-s + (−0.541 + 0.197i)3-s + (0.173 + 0.984i)4-s + (0.173 − 0.984i)5-s + (0.541 + 0.197i)6-s + (2.43 + 4.21i)7-s + (0.500 − 0.866i)8-s + (−2.04 + 1.71i)9-s + (−0.766 + 0.642i)10-s + (2.68 − 4.64i)11-s + (−0.288 − 0.499i)12-s + (3.62 + 1.31i)13-s + (0.844 − 4.79i)14-s + (0.100 + 0.567i)15-s + (−0.939 + 0.342i)16-s + (1.07 + 0.901i)17-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (−0.312 + 0.113i)3-s + (0.0868 + 0.492i)4-s + (0.0776 − 0.440i)5-s + (0.221 + 0.0804i)6-s + (0.919 + 1.59i)7-s + (0.176 − 0.306i)8-s + (−0.681 + 0.571i)9-s + (−0.242 + 0.203i)10-s + (0.809 − 1.40i)11-s + (−0.0831 − 0.144i)12-s + (1.00 + 0.365i)13-s + (0.225 − 1.28i)14-s + (0.0258 + 0.146i)15-s + (−0.234 + 0.0855i)16-s + (0.260 + 0.218i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $0.997 - 0.0671i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{190} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 190,\ (\ :1/2),\ 0.997 - 0.0671i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.922828 + 0.0310338i\)
\(L(\frac12)\) \(\approx\) \(0.922828 + 0.0310338i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (-4.35 + 0.226i)T \)
good3 \( 1 + (0.541 - 0.197i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-2.43 - 4.21i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.68 + 4.64i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.62 - 1.31i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.07 - 0.901i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.927 - 5.25i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (2.78 - 2.33i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (4.10 + 7.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 + (-1.79 + 0.652i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.256 + 1.45i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (2.39 - 2.00i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (0.312 + 1.77i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (5.61 + 4.70i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.40 - 7.99i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-6.46 + 5.42i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.04 + 5.93i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-1.19 + 0.436i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-15.6 + 5.68i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (7.23 + 12.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.02 + 3.28i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (4.65 + 3.90i)T + (16.8 + 95.5i)T^{2} \)
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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

−12.10756970357093362379447383656, −11.43348938348962959758222285795, −11.03548589706955339535689442075, −9.248084548007150535837542430317, −8.763359858151728520234251961420, −7.933223226435744206774218002023, −5.95484317344227868052988478228, −5.30569960067017941245904813418, −3.40924458962546139149572900868, −1.68757290958890336619381577458, 1.27946738305926572915466329657, 3.76794698874754528655155577042, 5.17243027402161210949226476946, 6.67306152057341175292245463683, 7.24347371535944515307559482891, 8.386373789764216950626320487157, 9.623546660637370482829511700993, 10.61522228913019980839473637761, 11.28531139384974613653763298578, 12.39737812534689824884952278055

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