↑

Properties

Label 2-189-21.17-c1-0-5
Degree $2$
Conductor $189$
Sign $0.997 + 0.0633i$
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  + (1.22 − 0.707i)2-s + (1.22 + 2.12i)5-s + (2.5 − 0.866i)7-s + 2.82i·8-s + (3 + 1.73i)10-s + (−4.89 − 2.82i)11-s − 3.46i·13-s + (2.44 − 2.82i)14-s + (2.00 + 3.46i)16-s + (1.22 − 2.12i)17-s + (1.5 − 0.866i)19-s − 8·22-s + (−6.12 + 3.53i)23-s + (−0.499 + 0.866i)25-s + (−2.44 − 4.24i)26-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)2-s + (0.547 + 0.948i)5-s + (0.944 − 0.327i)7-s + 0.999i·8-s + (0.948 + 0.547i)10-s + (−1.47 − 0.852i)11-s − 0.960i·13-s + (0.654 − 0.755i)14-s + (0.500 + 0.866i)16-s + (0.297 − 0.514i)17-s + (0.344 − 0.198i)19-s − 1.70·22-s + (−1.27 + 0.737i)23-s + (−0.0999 + 0.173i)25-s + (−0.480 − 0.832i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.997 + 0.0633i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.997 + 0.0633i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84164 - 0.0583769i\)
\(L(\frac12)\) \(\approx\) \(1.84164 - 0.0583769i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 0.866i)T \)
good2 \( 1 + (-1.22 + 0.707i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.22 - 2.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.89 + 2.82i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (-1.22 + 2.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.5 + 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.12 - 3.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 + (4.5 + 2.59i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.44T + 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 + (-3.67 - 6.36i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.22 + 0.707i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.34 + 12.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.82iT - 71T^{2} \)
73 \( 1 + (-10.5 - 6.06i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.79T + 83T^{2} \)
89 \( 1 + (1.22 + 2.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.66iT - 97T^{2} \)
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

−12.72661677005300829742858186927, −11.46164323713634425751724908063, −10.87698903361428098748662922302, −10.00645608608081351561977265559, −8.238439029753145795952760410200, −7.59746714390585447706212114441, −5.80396961269920167739436713063, −5.05064475941037451959452346539, −3.43660598060250442539838764466, −2.43187754419772588526089620704, 1.90748579708039254720588099870, 4.28653418143323724872235954554, 5.11314226612979465325326970094, 5.83485878016172540859065191283, 7.33341581612845252273168719148, 8.463681845534553039326776225981, 9.601338032656928756492484740107, 10.51806291731875016993377065202, 12.04998779011347332269161822310, 12.73760991071333305986867256581

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