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Properties

Label 2-189-63.58-c1-0-2
Degree $2$
Conductor $189$
Sign $-0.0907 + 0.995i$
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  − 2.05·2-s + 2.21·4-s + (−0.0731 − 0.126i)5-s + (−2.33 − 1.25i)7-s − 0.446·8-s + (0.150 + 0.260i)10-s + (0.832 − 1.44i)11-s + (0.0999 − 0.173i)13-s + (4.78 + 2.57i)14-s − 3.51·16-s + (−3.13 − 5.43i)17-s + (3.45 − 5.99i)19-s + (−0.162 − 0.280i)20-s + (−1.70 + 2.95i)22-s + (−3.09 − 5.35i)23-s + ⋯
L(s)  = 1  − 1.45·2-s + 1.10·4-s + (−0.0327 − 0.0566i)5-s + (−0.880 − 0.473i)7-s − 0.157·8-s + (0.0474 + 0.0822i)10-s + (0.250 − 0.434i)11-s + (0.0277 − 0.0480i)13-s + (1.27 + 0.687i)14-s − 0.879·16-s + (−0.760 − 1.31i)17-s + (0.793 − 1.37i)19-s + (−0.0362 − 0.0627i)20-s + (−0.364 + 0.630i)22-s + (−0.644 − 1.11i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0907 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0907 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.0907 + 0.995i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.0907 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.278699 - 0.305252i\)
\(L(\frac12)\) \(\approx\) \(0.278699 - 0.305252i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.33 + 1.25i)T \)
good2 \( 1 + 2.05T + 2T^{2} \)
5 \( 1 + (0.0731 + 0.126i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.832 + 1.44i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.0999 + 0.173i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.13 + 5.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.45 + 5.99i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.09 + 5.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.46 - 4.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.51T + 31T^{2} \)
37 \( 1 + (3.50 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.15 - 2.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.940 + 1.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.81T + 47T^{2} \)
53 \( 1 + (-2.67 - 4.62i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.57T + 59T^{2} \)
61 \( 1 + 0.678T + 61T^{2} \)
67 \( 1 + 6.18T + 67T^{2} \)
71 \( 1 + 1.27T + 71T^{2} \)
73 \( 1 + (0.778 + 1.34i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + (3.75 + 6.50i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.53 - 7.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.98 + 6.90i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.01458343930925282942502020714, −10.98287715000688767761833131441, −10.16206432534117705805037559699, −9.236770611653497375861339564687, −8.566967883270899183224880879857, −7.21441054481568655173208502662, −6.58510268601240302265824110023, −4.67056159929301481952571303676, −2.80914277246483510031802057840, −0.59034995183602076093628037285, 1.84585434494345503365220949432, 3.76889846405000270772698489281, 5.76832140419754963255405264580, 6.92670929414201911732642131394, 7.957514540451006023962298670009, 8.968460442106575147625646726242, 9.736779513399297199120935595802, 10.48614555714551117394517028652, 11.63898434530533670212787533890, 12.62362434516931202091723302524

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