Properties

Label 2-189-63.41-c1-0-1
Degree $2$
Conductor $189$
Sign $0.273 - 0.961i$
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  + (0.555 − 0.320i)2-s + (−0.794 + 1.37i)4-s + (−1.10 + 1.91i)5-s + (−0.906 + 2.48i)7-s + 2.30i·8-s + 1.41i·10-s + (2.93 − 1.69i)11-s + (−1.56 − 0.901i)13-s + (0.293 + 1.67i)14-s + (−0.849 − 1.47i)16-s + 5.96·17-s − 1.64i·19-s + (−1.75 − 3.04i)20-s + (1.08 − 1.88i)22-s + (−2.05 − 1.18i)23-s + ⋯
L(s)  = 1  + (0.392 − 0.226i)2-s + (−0.397 + 0.687i)4-s + (−0.494 + 0.856i)5-s + (−0.342 + 0.939i)7-s + 0.813i·8-s + 0.448i·10-s + (0.885 − 0.511i)11-s + (−0.432 − 0.249i)13-s + (0.0785 + 0.446i)14-s + (−0.212 − 0.367i)16-s + 1.44·17-s − 0.377i·19-s + (−0.392 − 0.680i)20-s + (0.232 − 0.401i)22-s + (−0.428 − 0.247i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.273 - 0.961i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.273 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.924192 + 0.698264i\)
\(L(\frac12)\) \(\approx\) \(0.924192 + 0.698264i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.906 - 2.48i)T \)
good2 \( 1 + (-0.555 + 0.320i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.10 - 1.91i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.93 + 1.69i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.56 + 0.901i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.96T + 17T^{2} \)
19 \( 1 + 1.64iT - 19T^{2} \)
23 \( 1 + (2.05 + 1.18i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.44 - 1.41i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-9.28 - 5.36i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.69T + 37T^{2} \)
41 \( 1 + (-0.455 + 0.788i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.96 + 3.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.123 + 0.213i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.87iT - 53T^{2} \)
59 \( 1 + (-5.39 + 9.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.22 + 0.709i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.99 + 6.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 - 0.426iT - 73T^{2} \)
79 \( 1 + (-2.49 - 4.31i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.28 - 7.42i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 + (6.30 - 3.63i)T + (48.5 - 84.0i)T^{2} \)
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

−12.44308102851271318767171609829, −12.02856131383925434727241785237, −11.11258082527295330491093762983, −9.777667697227297575021888323717, −8.689964715674137220625926729322, −7.74265179785656500425525957628, −6.50884285892603977681545242420, −5.18998236760881071025026278186, −3.64138506773329048308771201641, −2.83564868048947459662711409844, 1.03846462802600001614286503433, 3.88902011780119719378323057258, 4.61884076897991055353850629959, 5.93029425062456273181453354954, 7.10983858372584858229011754046, 8.261751789622790835864966078624, 9.689235478620630187341043701245, 10.01218485472795640995063993534, 11.62293045494175763284716762199, 12.47489133484027218838111942785

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