Properties

Label 2-189-27.16-c1-0-7
Degree $2$
Conductor $189$
Sign $0.211 - 0.977i$
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  + (1.89 + 1.58i)2-s + (−1.45 − 0.942i)3-s + (0.711 + 4.03i)4-s + (2.16 + 0.788i)5-s + (−1.25 − 4.08i)6-s + (0.173 − 0.984i)7-s + (−2.59 + 4.48i)8-s + (1.22 + 2.73i)9-s + (2.84 + 4.93i)10-s + (−4.00 + 1.45i)11-s + (2.76 − 6.53i)12-s + (1.41 − 1.18i)13-s + (1.89 − 1.58i)14-s + (−2.40 − 3.18i)15-s + (−4.32 + 1.57i)16-s + (−2.58 − 4.47i)17-s + ⋯
L(s)  = 1  + (1.33 + 1.12i)2-s + (−0.839 − 0.544i)3-s + (0.355 + 2.01i)4-s + (0.968 + 0.352i)5-s + (−0.511 − 1.66i)6-s + (0.0656 − 0.372i)7-s + (−0.916 + 1.58i)8-s + (0.407 + 0.913i)9-s + (0.900 + 1.55i)10-s + (−1.20 + 0.439i)11-s + (0.799 − 1.88i)12-s + (0.392 − 0.329i)13-s + (0.505 − 0.424i)14-s + (−0.620 − 0.823i)15-s + (−1.08 + 0.393i)16-s + (−0.626 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.211 - 0.977i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.211 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52599 + 1.23076i\)
\(L(\frac12)\) \(\approx\) \(1.52599 + 1.23076i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.45 + 0.942i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
good2 \( 1 + (-1.89 - 1.58i)T + (0.347 + 1.96i)T^{2} \)
5 \( 1 + (-2.16 - 0.788i)T + (3.83 + 3.21i)T^{2} \)
11 \( 1 + (4.00 - 1.45i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-1.41 + 1.18i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (2.58 + 4.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.742 + 1.28i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.717 + 4.06i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-7.51 - 6.30i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.813 + 4.61i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (3.72 + 6.44i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.88 - 5.77i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (5.92 - 2.15i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.69 - 9.59i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 + (8.33 + 3.03i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.844 + 4.79i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-0.364 + 0.306i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-4.51 - 7.82i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.54 - 9.60i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.28 - 2.75i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-4.66 - 3.91i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (5.05 - 8.76i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.34 + 1.21i)T + (74.3 - 62.3i)T^{2} \)
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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

−13.11285520620257786697993870447, −12.28284953606816971687498800479, −11.02943230919845995498246811589, −10.06022996915000625956969313896, −8.146473394515260607277823383869, −7.08964673371214405805455249816, −6.44376273397509153558795001047, −5.39301895658166276265663164004, −4.68654580654164637354001501403, −2.64786502076228325653841775486, 1.82751247942138018953648542280, 3.47208859490100430762661419725, 4.83454175328636036548745981722, 5.56727921706633783129028145107, 6.32520740831552801173179803312, 8.692114110750666703820175269935, 10.16498660623010200655959038699, 10.41180499777114597496600096687, 11.60809514612948760471219217049, 12.23987240967692287712154373257

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