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Properties

Label 2-189-189.25-c1-0-21
Degree $2$
Conductor $189$
Sign $0.245 + 0.969i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.90 − 0.694i)2-s + (0.482 − 1.66i)3-s + (1.62 − 1.36i)4-s + (−2.38 − 0.869i)5-s + (−0.234 − 3.50i)6-s + (1.31 + 2.29i)7-s + (0.123 − 0.213i)8-s + (−2.53 − 1.60i)9-s − 5.16·10-s + (5.38 − 1.95i)11-s + (−1.48 − 3.36i)12-s + (0.494 + 2.80i)13-s + (4.09 + 3.47i)14-s + (−2.59 + 3.55i)15-s + (−0.649 + 3.68i)16-s − 4.83·17-s + ⋯
L(s)  = 1  + (1.34 − 0.490i)2-s + (0.278 − 0.960i)3-s + (0.812 − 0.681i)4-s + (−1.06 − 0.388i)5-s + (−0.0956 − 1.43i)6-s + (0.495 + 0.868i)7-s + (0.0436 − 0.0755i)8-s + (−0.844 − 0.535i)9-s − 1.63·10-s + (1.62 − 0.590i)11-s + (−0.428 − 0.970i)12-s + (0.137 + 0.778i)13-s + (1.09 + 0.928i)14-s + (−0.671 + 0.917i)15-s + (−0.162 + 0.921i)16-s − 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.245 + 0.969i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.245 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70397 - 1.32651i\)
\(L(\frac12)\) \(\approx\) \(1.70397 - 1.32651i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.482 + 1.66i)T \)
7 \( 1 + (-1.31 - 2.29i)T \)
good2 \( 1 + (-1.90 + 0.694i)T + (1.53 - 1.28i)T^{2} \)
5 \( 1 + (2.38 + 0.869i)T + (3.83 + 3.21i)T^{2} \)
11 \( 1 + (-5.38 + 1.95i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.494 - 2.80i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + 4.83T + 17T^{2} \)
19 \( 1 - 0.555T + 19T^{2} \)
23 \( 1 + (-0.476 - 2.70i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-0.164 + 0.930i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-3.82 + 3.21i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (4.08 - 7.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.92 + 10.8i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (4.80 + 4.02i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (2.28 + 1.91i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-3.07 + 5.32i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.64 - 9.35i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-3.92 - 3.29i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (8.94 + 3.25i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (5.28 + 9.14i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.228 - 0.395i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.09 - 1.12i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-0.441 + 2.50i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + 6.69T + 89T^{2} \)
97 \( 1 + (-4.52 - 3.79i)T + (16.8 + 95.5i)T^{2} \)
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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

−12.05040299306327826503468592105, −11.85609819376405666645496107624, −11.33225396126458689264373719431, −8.882970888133247302491556599912, −8.497344391116344536832505624763, −6.91096734823655758456438104033, −5.91386265033323669319296990768, −4.48475884298488895982518132729, −3.47765572560985056394922534270, −1.88745083177167724371497057877, 3.32742775175925503497421308504, 4.17270030431969160833807449456, 4.77319535444952579551929991055, 6.43548283920838988020238271339, 7.38291090079255821275888202422, 8.610386589908810998982820314160, 9.943115592675541233980925188131, 11.13918483880886157466599431612, 11.75194738711534815822474148739, 12.98447359064410680898328250150

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