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Properties

Label 2-189-189.5-c1-0-9
Degree $2$
Conductor $189$
Sign $0.908 + 0.417i$
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  + (−2.31 + 0.408i)2-s + (−0.111 − 1.72i)3-s + (3.32 − 1.21i)4-s + (−0.175 + 0.996i)5-s + (0.965 + 3.96i)6-s + (1.63 + 2.08i)7-s + (−3.14 + 1.81i)8-s + (−2.97 + 0.386i)9-s − 2.38i·10-s + (0.972 − 0.171i)11-s + (−2.46 − 5.61i)12-s + (3.55 − 4.23i)13-s + (−4.63 − 4.15i)14-s + (1.74 + 0.192i)15-s + (1.11 − 0.936i)16-s + 3.54·17-s + ⋯
L(s)  = 1  + (−1.63 + 0.289i)2-s + (−0.0645 − 0.997i)3-s + (1.66 − 0.605i)4-s + (−0.0786 + 0.445i)5-s + (0.394 + 1.61i)6-s + (0.617 + 0.786i)7-s + (−1.11 + 0.641i)8-s + (−0.991 + 0.128i)9-s − 0.753i·10-s + (0.293 − 0.0517i)11-s + (−0.711 − 1.62i)12-s + (0.984 − 1.17i)13-s + (−1.24 − 1.11i)14-s + (0.449 + 0.0496i)15-s + (0.278 − 0.234i)16-s + 0.859·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.908 + 0.417i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.908 + 0.417i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.576319 - 0.125982i\)
\(L(\frac12)\) \(\approx\) \(0.576319 - 0.125982i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.111 + 1.72i)T \)
7 \( 1 + (-1.63 - 2.08i)T \)
good2 \( 1 + (2.31 - 0.408i)T + (1.87 - 0.684i)T^{2} \)
5 \( 1 + (0.175 - 0.996i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-0.972 + 0.171i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (-3.55 + 4.23i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 - 3.54T + 17T^{2} \)
19 \( 1 + 0.366iT - 19T^{2} \)
23 \( 1 + (-4.11 + 4.90i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (0.703 + 0.838i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (2.36 + 6.49i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (-2.23 - 3.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.350 - 0.293i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (9.94 + 3.61i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-10.6 - 3.88i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (6.67 - 3.85i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.199 + 0.167i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (3.45 - 9.48i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.0909 - 0.516i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (4.17 + 2.40i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-9.88 - 5.70i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.918 - 5.20i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-7.16 + 6.01i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 - 2.53T + 89T^{2} \)
97 \( 1 + (-0.637 + 1.75i)T + (-74.3 - 62.3i)T^{2} \)
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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

−12.23804825751724229836945712721, −11.20340798651378888246747303964, −10.60309478450283201257451189294, −9.163149359534601883450877901877, −8.336587287692901865389622734158, −7.70431211605259227174964295773, −6.60480864970594307409731251964, −5.63595234018310637298600324908, −2.75459192840020696569760536837, −1.15864597934355032264949371091, 1.35543762461597043874054444677, 3.60411108789372258917652614797, 4.99036081968854192252652025461, 6.77339378088091581357417400187, 8.001081476349982912324155501349, 8.895206194823295184048344235959, 9.513963362028443864078019844391, 10.62738022211063281561618048863, 11.14567392000419807217044264127, 12.02326427447476773726285795973

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