Properties

Label 2-190-19.4-c1-0-7
Degree $2$
Conductor $190$
Sign $0.235 + 0.971i$
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 + (−0.558 − 3.16i)3-s + (0.766 + 0.642i)4-s + (0.766 − 0.642i)5-s + (0.558 − 3.16i)6-s + (0.0116 − 0.0202i)7-s + (0.500 + 0.866i)8-s + (−6.89 + 2.51i)9-s + (0.939 − 0.342i)10-s + (−1.08 − 1.87i)11-s + (1.60 − 2.78i)12-s + (−0.276 + 1.56i)13-s + (0.0179 − 0.0150i)14-s + (−2.46 − 2.06i)15-s + (0.173 + 0.984i)16-s + (7.49 + 2.72i)17-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (−0.322 − 1.82i)3-s + (0.383 + 0.321i)4-s + (0.342 − 0.287i)5-s + (0.227 − 1.29i)6-s + (0.00442 − 0.00765i)7-s + (0.176 + 0.306i)8-s + (−2.29 + 0.836i)9-s + (0.297 − 0.108i)10-s + (−0.325 − 0.564i)11-s + (0.464 − 0.803i)12-s + (−0.0765 + 0.434i)13-s + (0.00478 − 0.00401i)14-s + (−0.636 − 0.533i)15-s + (0.0434 + 0.246i)16-s + (1.81 + 0.661i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $0.235 + 0.971i$
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: \((2,\ 190,\ (\ :1/2),\ 0.235 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22095 - 0.960266i\)
\(L(\frac12)\) \(\approx\) \(1.22095 - 0.960266i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.06 + 4.22i)T \)
good3 \( 1 + (0.558 + 3.16i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (-0.0116 + 0.0202i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.08 + 1.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.276 - 1.56i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-7.49 - 2.72i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-3.43 - 2.88i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (7.09 - 2.58i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (2.22 - 3.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.389T + 37T^{2} \)
41 \( 1 + (-0.972 - 5.51i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-7.43 + 6.23i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (4.01 - 1.46i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (2.89 + 2.43i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-3.41 - 1.24i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (8.84 + 7.41i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (3.62 - 1.31i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (9.85 - 8.27i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.330 + 1.87i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (1.65 + 9.36i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-1.31 + 2.27i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.54 + 8.74i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (10.4 + 3.81i)T + (74.3 + 62.3i)T^{2} \)
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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

−12.61143355836178911918931654479, −11.75524516009922058033241462108, −10.85773763566832484602121808056, −9.031930978365775595658178582241, −7.83089003174240823550031975699, −7.14807799854427395880965654405, −5.98660439237550053426189062065, −5.30829184557074324958913404097, −3.06420170532237962167805788610, −1.45513449057481048142848604328, 2.95371432639306828263201144443, 4.01052783924476021343912803386, 5.24882316009040232360915824759, 5.81687406050195603012543903052, 7.63940053569355466404016394059, 9.329110423072488254430038023115, 10.01496960548138457804207449532, 10.66635988990291927735530107585, 11.63935845648922816091608050199, 12.60009826353248086658153214265

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