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Properties

Label 2-189-189.5-c1-0-7
Degree $2$
Conductor $189$
Sign $0.805 + 0.592i$
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  + (−1.21 + 0.213i)2-s + (−1.60 − 0.659i)3-s + (−0.452 + 0.164i)4-s + (−0.378 + 2.14i)5-s + (2.08 + 0.457i)6-s + (−0.671 − 2.55i)7-s + (2.64 − 1.52i)8-s + (2.13 + 2.11i)9-s − 2.68i·10-s + (−0.0432 + 0.00762i)11-s + (0.833 + 0.0346i)12-s + (2.80 − 3.34i)13-s + (1.36 + 2.96i)14-s + (2.01 − 3.18i)15-s + (−2.14 + 1.80i)16-s + 4.53·17-s + ⋯
L(s)  = 1  + (−0.858 + 0.151i)2-s + (−0.924 − 0.380i)3-s + (−0.226 + 0.0823i)4-s + (−0.169 + 0.959i)5-s + (0.851 + 0.186i)6-s + (−0.253 − 0.967i)7-s + (0.936 − 0.540i)8-s + (0.710 + 0.704i)9-s − 0.848i·10-s + (−0.0130 + 0.00229i)11-s + (0.240 + 0.00999i)12-s + (0.779 − 0.928i)13-s + (0.364 + 0.791i)14-s + (0.521 − 0.822i)15-s + (−0.537 + 0.450i)16-s + 1.10·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.805 + 0.592i$
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: \((2,\ 189,\ (\ :1/2),\ 0.805 + 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.456728 - 0.149987i\)
\(L(\frac12)\) \(\approx\) \(0.456728 - 0.149987i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.60 + 0.659i)T \)
7 \( 1 + (0.671 + 2.55i)T \)
good2 \( 1 + (1.21 - 0.213i)T + (1.87 - 0.684i)T^{2} \)
5 \( 1 + (0.378 - 2.14i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (0.0432 - 0.00762i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (-2.80 + 3.34i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 - 4.53T + 17T^{2} \)
19 \( 1 + 6.99iT - 19T^{2} \)
23 \( 1 + (-0.167 + 0.199i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-2.65 - 3.16i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.31 - 3.61i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (2.18 + 3.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.10 + 4.28i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-9.15 - 3.33i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (6.09 + 2.21i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-7.32 + 4.23i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.98 + 6.70i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-3.82 + 10.5i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.622 + 3.52i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-5.14 - 2.97i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.45 - 3.14i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.77 - 10.0i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-9.64 + 8.08i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 - 0.332T + 89T^{2} \)
97 \( 1 + (1.19 - 3.27i)T + (-74.3 - 62.3i)T^{2} \)
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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

−12.49764026003567058053321840704, −10.93254632978513625492618653069, −10.69646843137186209003470110223, −9.707157701848171769837619898971, −8.215198484503598395351427801426, −7.25532826637636660276557658687, −6.64543209684670514056403126456, −5.04062252669787682636656188718, −3.48656348879277899310806588082, −0.794176763296737058812679502754, 1.30074707709714699094468160323, 4.08975201425832343953031325458, 5.24708568080749416625769171774, 6.16810407254836493527012609979, 7.958816930707060857405386039710, 8.875016450111249477675351589707, 9.643240174623423232794038478196, 10.48197733442085379905234046595, 11.77929876278553001418934087882, 12.24118367051338197046740831614

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