Properties

Label 2-190-19.4-c1-0-4
Degree $2$
Conductor $190$
Sign $0.914 - 0.404i$
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  + (0.939 + 0.342i)2-s + (0.00627 + 0.0355i)3-s + (0.766 + 0.642i)4-s + (0.766 − 0.642i)5-s + (−0.00627 + 0.0355i)6-s + (−0.918 + 1.59i)7-s + (0.500 + 0.866i)8-s + (2.81 − 1.02i)9-s + (0.939 − 0.342i)10-s + (1.23 + 2.13i)11-s + (−0.0180 + 0.0313i)12-s + (0.415 − 2.35i)13-s + (−1.40 + 1.18i)14-s + (0.0276 + 0.0232i)15-s + (0.173 + 0.984i)16-s + (−6.33 − 2.30i)17-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (0.00362 + 0.0205i)3-s + (0.383 + 0.321i)4-s + (0.342 − 0.287i)5-s + (−0.00256 + 0.0145i)6-s + (−0.347 + 0.601i)7-s + (0.176 + 0.306i)8-s + (0.939 − 0.341i)9-s + (0.297 − 0.108i)10-s + (0.371 + 0.643i)11-s + (−0.00521 + 0.00903i)12-s + (0.115 − 0.653i)13-s + (−0.376 + 0.315i)14-s + (0.00714 + 0.00599i)15-s + (0.0434 + 0.246i)16-s + (−1.53 − 0.559i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $0.914 - 0.404i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{190} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 190,\ (\ :1/2),\ 0.914 - 0.404i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70967 + 0.360943i\)
\(L(\frac12)\) \(\approx\) \(1.70967 + 0.360943i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (4.34 + 0.298i)T \)
good3 \( 1 + (-0.00627 - 0.0355i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (0.918 - 1.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.23 - 2.13i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.415 + 2.35i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (6.33 + 2.30i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-1.24 - 1.04i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (3.10 - 1.12i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-1.75 + 3.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.00T + 37T^{2} \)
41 \( 1 + (1.38 + 7.85i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-4.37 + 3.67i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (11.7 - 4.27i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (1.47 + 1.23i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-4.46 - 1.62i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-10.4 - 8.78i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-3.49 + 1.27i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-4.46 + 3.74i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.0886 + 0.502i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-2.10 - 11.9i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (3.11 - 5.39i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.95 + 16.7i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-12.8 - 4.67i)T + (74.3 + 62.3i)T^{2} \)
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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

−12.84774534041614501833266443616, −11.95465359855730984569660252024, −10.73317496289173537960528121243, −9.570352084890643088411577697125, −8.682364468009086539369136525306, −7.16752264243759248936771552918, −6.35164136597796450089010182407, −5.07507508346999410522835872311, −3.96462896839289566046761946984, −2.21554340582056243304813703371, 1.93479151852949990431148058137, 3.70952112232813029665453131201, 4.69635543657869100849956282847, 6.39059633144509163395666522404, 6.88256089656427363948076783629, 8.500512646740704049052205288010, 9.779745979845375529601880419742, 10.70043612954315243659122220819, 11.40345405643523160900147823798, 12.85679045592544362968700159995

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