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Properties

Label 2-189-21.17-c1-0-10
Degree $2$
Conductor $189$
Sign $0.244 + 0.969i$
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  + (1.58 − 0.917i)2-s + (0.682 − 1.18i)4-s + (−1.90 − 3.29i)5-s + (2.11 − 1.58i)7-s + 1.16i·8-s + (−6.04 − 3.49i)10-s + (1.00 + 0.582i)11-s + 5.54i·13-s + (1.90 − 4.46i)14-s + (2.43 + 4.21i)16-s + (1.58 − 2.75i)17-s + (−0.546 + 0.315i)19-s − 5.19·20-s + 2.13·22-s + (−1.52 + 0.880i)23-s + ⋯
L(s)  = 1  + (1.12 − 0.648i)2-s + (0.341 − 0.590i)4-s + (−0.851 − 1.47i)5-s + (0.799 − 0.600i)7-s + 0.412i·8-s + (−1.91 − 1.10i)10-s + (0.304 + 0.175i)11-s + 1.53i·13-s + (0.508 − 1.19i)14-s + (0.608 + 1.05i)16-s + (0.385 − 0.667i)17-s + (−0.125 + 0.0724i)19-s − 1.16·20-s + 0.455·22-s + (−0.318 + 0.183i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.244 + 0.969i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.244 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47613 - 1.14995i\)
\(L(\frac12)\) \(\approx\) \(1.47613 - 1.14995i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.11 + 1.58i)T \)
good2 \( 1 + (-1.58 + 0.917i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.90 + 3.29i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.00 - 0.582i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.54iT - 13T^{2} \)
17 \( 1 + (-1.58 + 2.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.546 - 0.315i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.52 - 0.880i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.83iT - 29T^{2} \)
31 \( 1 + (3.70 + 2.13i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.11 + 3.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.56T + 41T^{2} \)
43 \( 1 - 7.23T + 43T^{2} \)
47 \( 1 + (1.27 + 2.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0627 - 0.0362i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.49 - 6.04i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.00 + 4.04i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.54 - 4.41i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.76iT - 71T^{2} \)
73 \( 1 + (-4.84 - 2.79i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.54 + 14.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + (-5.77 - 10.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.688iT - 97T^{2} \)
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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

−12.19275218204896694933437796099, −11.79303635901013486029361482978, −10.94539665148817168489856545570, −9.272786167368186191706565457624, −8.371310953175363849939441258240, −7.25863314144567465941719138208, −5.34873161510686683829534024466, −4.48149939732343399772445372764, −3.86599776265850199085720843752, −1.63649846551935434181554606907, 2.96794080793941839423489212343, 4.01017293493023535629077665395, 5.43096866486366871921976203552, 6.34700139638542543082677709435, 7.47628633510653001601674173263, 8.245634178668131832678468101777, 10.10563375861444202699405975483, 10.99881195876584158481961985532, 11.95732362607597373393980263911, 12.83776050303653185253128180420

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