Properties

Label 2-190-19.6-c1-0-1
Degree $2$
Conductor $190$
Sign $0.264 - 0.964i$
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 + (2.22 + 0.811i)3-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (−2.22 + 0.811i)6-s + (−1.57 + 2.73i)7-s + (0.500 + 0.866i)8-s + (2.00 + 1.68i)9-s + (−0.766 − 0.642i)10-s + (−0.688 − 1.19i)11-s + (1.18 − 2.05i)12-s + (4.06 − 1.47i)13-s + (−0.547 − 3.10i)14-s + (−0.411 + 2.33i)15-s + (−0.939 − 0.342i)16-s + (−0.993 + 0.833i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (1.28 + 0.468i)3-s + (0.0868 − 0.492i)4-s + (0.0776 + 0.440i)5-s + (−0.909 + 0.331i)6-s + (−0.596 + 1.03i)7-s + (0.176 + 0.306i)8-s + (0.669 + 0.562i)9-s + (−0.242 − 0.203i)10-s + (−0.207 − 0.359i)11-s + (0.342 − 0.592i)12-s + (1.12 − 0.410i)13-s + (−0.146 − 0.830i)14-s + (−0.106 + 0.602i)15-s + (−0.234 − 0.0855i)16-s + (−0.240 + 0.202i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $0.264 - 0.964i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{190} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 190,\ (\ :1/2),\ 0.264 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02308 + 0.780271i\)
\(L(\frac12)\) \(\approx\) \(1.02308 + 0.780271i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.08 + 4.22i)T \)
good3 \( 1 + (-2.22 - 0.811i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (1.57 - 2.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.688 + 1.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.06 + 1.47i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.993 - 0.833i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (0.369 - 2.09i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (0.0998 + 0.0837i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.173 - 0.300i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.3T + 37T^{2} \)
41 \( 1 + (10.3 + 3.76i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (2.00 + 11.3i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-0.0585 - 0.0491i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-1.08 + 6.12i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (6.27 - 5.26i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.36 + 7.72i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (0.789 + 0.662i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.02 - 11.4i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-11.5 - 4.18i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (13.1 + 4.79i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (5.68 - 9.85i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-17.1 + 6.23i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (12.4 - 10.4i)T + (16.8 - 95.5i)T^{2} \)
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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

−13.14482218625273110516745711319, −11.58414137219503226170153013407, −10.46844417170843434428926566329, −9.442069683797859126627549771155, −8.795454116897007075933615901930, −8.042757541769009150680843746978, −6.64637133615044384339441840193, −5.53406177719607673492853812023, −3.58785302087501228870285377368, −2.49896957647338591792666511834, 1.49522190374989653156568944878, 3.09797616203591873879154278088, 4.20830825846434393385191762423, 6.43987302792540986678398717073, 7.61722741107381077511211732274, 8.337628661285927640435933743737, 9.330014769711474431157624986114, 10.11575281475099341648920487182, 11.28844509473031360665311263462, 12.60545768587124181541100750354

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