↑

Properties

Label 2-189-63.23-c2-0-11
Degree $2$
Conductor $189$
Sign $-0.367 + 0.929i$
Analytic cond. $5.14987$
Root an. cond. $2.26933$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.46i·2-s + 1.85·4-s + (−0.998 + 0.576i)5-s + (−4.05 − 5.70i)7-s − 8.57i·8-s + (0.844 + 1.46i)10-s + (−0.209 − 0.120i)11-s + (7.70 − 13.3i)13-s + (−8.35 + 5.94i)14-s − 5.16·16-s + (10.9 − 6.29i)17-s + (13.7 − 23.7i)19-s + (−1.84 + 1.06i)20-s + (−0.176 + 0.306i)22-s + (−17.6 + 10.1i)23-s + ⋯
L(s)  = 1  − 0.732i·2-s + 0.463·4-s + (−0.199 + 0.115i)5-s + (−0.579 − 0.814i)7-s − 1.07i·8-s + (0.0844 + 0.146i)10-s + (−0.0190 − 0.0109i)11-s + (0.592 − 1.02i)13-s + (−0.596 + 0.424i)14-s − 0.322·16-s + (0.641 − 0.370i)17-s + (0.721 − 1.24i)19-s + (−0.0924 + 0.0533i)20-s + (−0.00803 + 0.0139i)22-s + (−0.767 + 0.443i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.367 + 0.929i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ -0.367 + 0.929i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.872914 - 1.28398i\)
\(L(\frac12)\) \(\approx\) \(0.872914 - 1.28398i\)
\(L(2)\) 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 \( 1 + (4.05 + 5.70i)T \)
good2 \( 1 + 1.46iT - 4T^{2} \)
5 \( 1 + (0.998 - 0.576i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (0.209 + 0.120i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-7.70 + 13.3i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-10.9 + 6.29i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-13.7 + 23.7i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (17.6 - 10.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (16.9 - 9.76i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + 2.72T + 961T^{2} \)
37 \( 1 + (11.4 - 19.8i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-61.9 - 35.7i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-14.8 - 25.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 29.9iT - 2.20e3T^{2} \)
53 \( 1 + (-90.5 + 52.3i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 - 86.7iT - 3.48e3T^{2} \)
61 \( 1 - 17.8T + 3.72e3T^{2} \)
67 \( 1 + 18.0T + 4.48e3T^{2} \)
71 \( 1 - 74.2iT - 5.04e3T^{2} \)
73 \( 1 + (-13.0 - 22.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 - 22.5T + 6.24e3T^{2} \)
83 \( 1 + (-77.0 + 44.5i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (49.7 + 28.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-73.9 - 128. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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

−11.84002775538518934927351661433, −11.08683686607461947407486880814, −10.22605624726853332171404596855, −9.413325581272459400508124581781, −7.74257304313452385515416549568, −6.98083684321909964728685886533, −5.66068706202974082979542023897, −3.83205409840024697008996179668, −2.92102148780207992632002212018, −0.945187285290497774872160146756, 2.13552560012133916296714279116, 3.83509376447292603470629717265, 5.65820345306766372349051402523, 6.23433776371449517809975538241, 7.49834637487127765524209156126, 8.417450858366064197923397555060, 9.490693317168566573854731867995, 10.70826085394020466332528044089, 11.91638107928481883258873312319, 12.34798829593692321935696964268

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