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Properties

Label 2-189-189.131-c1-0-18
Degree $2$
Conductor $189$
Sign $0.786 + 0.618i$
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  + (1.06 + 1.26i)2-s + (−0.257 − 1.71i)3-s + (−0.129 + 0.734i)4-s + (−2.93 − 2.46i)5-s + (1.90 − 2.15i)6-s + (2.02 − 1.70i)7-s + (1.79 − 1.03i)8-s + (−2.86 + 0.880i)9-s − 6.35i·10-s + (1.84 + 2.20i)11-s + (1.29 + 0.0330i)12-s + (1.44 + 3.96i)13-s + (4.31 + 0.750i)14-s + (−3.46 + 5.66i)15-s + (4.63 + 1.68i)16-s − 3.18·17-s + ⋯
L(s)  = 1  + (0.753 + 0.897i)2-s + (−0.148 − 0.988i)3-s + (−0.0647 + 0.367i)4-s + (−1.31 − 1.10i)5-s + (0.775 − 0.878i)6-s + (0.764 − 0.644i)7-s + (0.636 − 0.367i)8-s + (−0.955 + 0.293i)9-s − 2.01i·10-s + (0.557 + 0.663i)11-s + (0.372 + 0.00953i)12-s + (0.399 + 1.09i)13-s + (1.15 + 0.200i)14-s + (−0.895 + 1.46i)15-s + (1.15 + 0.422i)16-s − 0.773·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 + 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.786 + 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.786 + 0.618i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.786 + 0.618i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38874 - 0.480540i\)
\(L(\frac12)\) \(\approx\) \(1.38874 - 0.480540i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.257 + 1.71i)T \)
7 \( 1 + (-2.02 + 1.70i)T \)
good2 \( 1 + (-1.06 - 1.26i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (2.93 + 2.46i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (-1.84 - 2.20i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-1.44 - 3.96i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + 3.18T + 17T^{2} \)
19 \( 1 + 2.26iT - 19T^{2} \)
23 \( 1 + (0.782 + 2.14i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.356 + 0.979i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-8.54 - 1.50i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (0.0122 + 0.0211i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.99 + 1.09i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.110 + 0.627i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-1.86 - 10.5i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (1.91 - 1.10i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.2 - 3.74i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (7.50 - 1.32i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (8.32 + 6.98i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-3.88 - 2.24i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.92 - 1.11i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.68 - 8.13i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (4.16 + 1.51i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 - 4.73T + 89T^{2} \)
97 \( 1 + (-13.0 + 2.29i)T + (91.1 - 33.1i)T^{2} \)
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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

−12.56217736784913667912940166765, −11.81169099378735884813653579807, −10.95558433815234196981397823614, −8.965711486199324373539193638590, −7.986909555196555179754412164129, −7.23674230586698620984887750189, −6.35049055087739201199220477566, −4.68304890446335734983549466190, −4.31559983778055087465133865340, −1.29000546003868555407323154778, 2.86645384670826161467549723296, 3.67272317878507899907708022017, 4.65355184531080449305575011772, 6.02696003830911831853788018639, 7.80302587522496024542269548786, 8.582371837969418672991793083276, 10.32083285345731527647736994283, 11.02667365826392526870870113455, 11.56636185754032786056642637382, 12.17728965048922357788658240803

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