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Properties

Label 2-189-21.5-c1-0-10
Degree $2$
Conductor $189$
Sign $0.575 + 0.817i$
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  + (0.568 + 0.328i)2-s + (−0.784 − 1.35i)4-s + (1.65 − 2.86i)5-s + (−2.58 + 0.568i)7-s − 2.34i·8-s + (1.88 − 1.08i)10-s + (2.02 − 1.17i)11-s + 1.58i·13-s + (−1.65 − 0.524i)14-s + (−0.799 + 1.38i)16-s + (0.568 + 0.984i)17-s + (3.85 + 2.22i)19-s − 5.19·20-s + 1.53·22-s + (8.13 + 4.69i)23-s + ⋯
L(s)  = 1  + (0.402 + 0.232i)2-s + (−0.392 − 0.679i)4-s + (0.740 − 1.28i)5-s + (−0.976 + 0.214i)7-s − 0.828i·8-s + (0.595 − 0.343i)10-s + (0.611 − 0.353i)11-s + 0.438i·13-s + (−0.442 − 0.140i)14-s + (−0.199 + 0.346i)16-s + (0.137 + 0.238i)17-s + (0.884 + 0.510i)19-s − 1.16·20-s + 0.328·22-s + (1.69 + 0.979i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.575 + 0.817i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.575 + 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20488 - 0.625233i\)
\(L(\frac12)\) \(\approx\) \(1.20488 - 0.625233i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.58 - 0.568i)T \)
good2 \( 1 + (-0.568 - 0.328i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.65 + 2.86i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.02 + 1.17i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.58iT - 13T^{2} \)
17 \( 1 + (-0.568 - 0.984i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.85 - 2.22i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-8.13 - 4.69i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.65iT - 29T^{2} \)
31 \( 1 + (6.33 - 3.65i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.58 + 4.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.64T + 41T^{2} \)
43 \( 1 + 2.16T + 43T^{2} \)
47 \( 1 + (2.79 - 4.83i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.70 + 5.02i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.08 - 1.88i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.46 - 2.00i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.38 - 9.32i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.39iT - 71T^{2} \)
73 \( 1 + (9.25 - 5.34i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.616 - 1.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.03T + 83T^{2} \)
89 \( 1 + (-3.73 + 6.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−12.83587594454581943392825794107, −11.64926807033391773527589133498, −10.10583612909667517834746960573, −9.332793364366319304133101240835, −8.838735311133403304916159826006, −6.92960193812229129517934150946, −5.79172972445034783562184033518, −5.15352584390238513922356248525, −3.69830741621834877958108922721, −1.28646409251738375091327508019, 2.73381909199278755860885802027, 3.51499434626570847547450702740, 5.17166649416870479313682152561, 6.62026284753832336757874602066, 7.28124101071169915718286818067, 8.914367053921505245292591462751, 9.809356224229891766186131171008, 10.78580810677736569879285630831, 11.80952552274336056431449746708, 12.94274596005260301966354215995

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