Properties

Label 2-190-19.16-c1-0-0
Degree $2$
Conductor $190$
Sign $-0.887 - 0.460i$
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 + (−1.68 + 0.613i)3-s + (0.173 + 0.984i)4-s + (0.173 − 0.984i)5-s + (1.68 + 0.613i)6-s + (−0.680 − 1.17i)7-s + (0.500 − 0.866i)8-s + (0.169 − 0.142i)9-s + (−0.766 + 0.642i)10-s + (−3.22 + 5.59i)11-s + (−0.897 − 1.55i)12-s + (−5.52 − 2.01i)13-s + (−0.236 + 1.34i)14-s + (0.311 + 1.76i)15-s + (−0.939 + 0.342i)16-s + (−1.96 − 1.64i)17-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (−0.973 + 0.354i)3-s + (0.0868 + 0.492i)4-s + (0.0776 − 0.440i)5-s + (0.688 + 0.250i)6-s + (−0.257 − 0.445i)7-s + (0.176 − 0.306i)8-s + (0.0566 − 0.0474i)9-s + (−0.242 + 0.203i)10-s + (−0.973 + 1.68i)11-s + (−0.259 − 0.448i)12-s + (−1.53 − 0.557i)13-s + (−0.0631 + 0.358i)14-s + (0.0804 + 0.456i)15-s + (−0.234 + 0.0855i)16-s + (−0.475 − 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 - 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $-0.887 - 0.460i$
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: \((2,\ 190,\ (\ :1/2),\ -0.887 - 0.460i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0138879 + 0.0569378i\)
\(L(\frac12)\) \(\approx\) \(0.0138879 + 0.0569378i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.83 + 2.06i)T \)
good3 \( 1 + (1.68 - 0.613i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (0.680 + 1.17i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.22 - 5.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.52 + 2.01i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.96 + 1.64i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-1.46 - 8.29i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-3.41 + 2.86i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.701 - 1.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.42T + 37T^{2} \)
41 \( 1 + (-5.15 + 1.87i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.19 + 6.74i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (7.29 - 6.11i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-0.768 - 4.35i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (7.87 + 6.60i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.12 - 6.36i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (5.14 - 4.31i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.336 + 1.90i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-5.90 + 2.14i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-3.37 + 1.22i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-6.74 - 11.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.78 - 0.648i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (0.0945 + 0.0793i)T + (16.8 + 95.5i)T^{2} \)
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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

−12.59585097269332206425490979262, −12.00325665475792607268132549633, −10.82192481827822368849542550886, −10.11296354089439109917572022907, −9.410462907433512241481625796284, −7.83038473045830132935656843684, −6.98710609765761069970388805330, −5.26177611459537978671360037812, −4.53931352452197599360557075049, −2.41477678417114575671769099627, 0.06374269368462409012132998672, 2.66691206729761745339915015627, 4.97451974634307921206844321834, 6.11733030965945305171884621825, 6.65887098325696669602905758677, 8.059986776735140634389607259031, 9.004283530469060126518668480362, 10.41422298142556033754654290522, 10.96636692033103173923877156371, 12.05977104587789440703278426838

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