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Properties

Label 2-189-7.4-c3-0-13
Degree $2$
Conductor $189$
Sign $-0.390 - 0.920i$
Analytic cond. $11.1513$
Root an. cond. $3.33936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.64 + 2.85i)2-s + (−1.42 + 2.46i)4-s + (10.8 + 18.8i)5-s + (1.08 − 18.4i)7-s + 16.9·8-s + (−35.7 + 61.9i)10-s + (−20.9 + 36.3i)11-s + 46.0·13-s + (54.5 − 27.3i)14-s + (39.3 + 68.1i)16-s + (1.00 − 1.74i)17-s + (−36.6 − 63.4i)19-s − 61.7·20-s − 138.·22-s + (−12.0 − 20.9i)23-s + ⋯
L(s)  = 1  + (0.582 + 1.00i)2-s + (−0.177 + 0.307i)4-s + (0.971 + 1.68i)5-s + (0.0587 − 0.998i)7-s + 0.750·8-s + (−1.13 + 1.95i)10-s + (−0.575 + 0.996i)11-s + 0.982·13-s + (1.04 − 0.521i)14-s + (0.614 + 1.06i)16-s + (0.0143 − 0.0249i)17-s + (−0.442 − 0.765i)19-s − 0.690·20-s − 1.33·22-s + (−0.109 − 0.189i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.390 - 0.920i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.390 - 0.920i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.61191 + 2.43543i\)
\(L(\frac12)\) \(\approx\) \(1.61191 + 2.43543i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.08 + 18.4i)T \)
good2 \( 1 + (-1.64 - 2.85i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-10.8 - 18.8i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (20.9 - 36.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 46.0T + 2.19e3T^{2} \)
17 \( 1 + (-1.00 + 1.74i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (36.6 + 63.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (12.0 + 20.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 90.7T + 2.43e4T^{2} \)
31 \( 1 + (26.6 - 46.0i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (33.9 + 58.7i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 341.T + 6.89e4T^{2} \)
43 \( 1 - 509.T + 7.95e4T^{2} \)
47 \( 1 + (-19.2 - 33.3i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-194. + 337. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-225. + 390. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (112. + 195. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-386. + 668. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 962.T + 3.57e5T^{2} \)
73 \( 1 + (-526. + 912. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (16.7 + 29.0i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 446.T + 5.71e5T^{2} \)
89 \( 1 + (-244. - 424. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 460.T + 9.12e5T^{2} \)
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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

−12.99533706810934044345771320377, −10.95175260533946955216196942357, −10.64688389287392793869866520073, −9.659705359884098091655228887057, −7.78314595729765964210155773229, −6.91640394367455827608764134946, −6.40958726670936811313925220147, −5.18204277843410194693642705519, −3.72550958859471944488408875859, −2.03516163479362097860128980115, 1.21734684374335654674535068697, 2.37387777120070897887747798619, 3.96687935494300154430483064295, 5.32087480781456467255366613973, 5.88148264887165944135521532748, 8.189507701983593713478633298312, 8.817762260113492249979880322250, 9.922742997312521327524901986688, 11.04422307749418819761644349102, 12.04412945231047885250578822936

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