↑

Properties

Label 2-189-63.4-c1-0-4
Degree $2$
Conductor $189$
Sign $0.975 + 0.220i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.500 + 0.866i)4-s + 5-s + (0.5 + 2.59i)7-s + 3·8-s + (0.5 − 0.866i)10-s − 5·11-s + (2.5 − 4.33i)13-s + (2.5 + 0.866i)14-s + (0.500 − 0.866i)16-s + (1.5 − 2.59i)17-s + (−0.5 − 0.866i)19-s + (0.500 + 0.866i)20-s + (−2.5 + 4.33i)22-s − 3·23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.250 + 0.433i)4-s + 0.447·5-s + (0.188 + 0.981i)7-s + 1.06·8-s + (0.158 − 0.273i)10-s − 1.50·11-s + (0.693 − 1.20i)13-s + (0.668 + 0.231i)14-s + (0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s + (−0.114 − 0.198i)19-s + (0.111 + 0.193i)20-s + (−0.533 + 0.923i)22-s − 0.625·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.975 + 0.220i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.975 + 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57795 - 0.176127i\)
\(L(\frac12)\) \(\approx\) \(1.57795 - 0.176127i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.5 - 2.59i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - T + 5T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (1.5 - 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.5 + 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.5 - 7.79i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.64440623041172584319540162341, −11.61049754995393916376844565011, −10.73552797197154587147109636488, −9.820741524510833736270812329267, −8.335743490566554719418023111861, −7.66094480501915480945253857803, −5.94431177626814035491798849008, −5.00571261370781038522520340926, −3.23360641549339548483224109574, −2.23682937701810946979391624099, 1.82747189575830799731420572209, 4.03357147087678414271010530640, 5.27745395674239555413144214031, 6.27681967613607538491015499824, 7.30876594798100087675587758522, 8.278093257197619904958850562194, 9.934778265525707550009118246164, 10.50724460328676014828734711336, 11.45864290406942219613481034488, 13.01002145692832672929896698278

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