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Properties

Label 2-189-21.20-c1-0-5
Degree $2$
Conductor $189$
Sign $0.981 + 0.188i$
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  + 2·4-s + (0.5 − 2.59i)7-s + 5.19i·13-s + 4·16-s − 5.19i·19-s − 5·25-s + (1 − 5.19i)28-s + 10.3i·31-s − 11·37-s − 8·43-s + (−6.5 − 2.59i)49-s + 10.3i·52-s − 15.5i·61-s + 8·64-s + 5·67-s + ⋯
L(s)  = 1  + 4-s + (0.188 − 0.981i)7-s + 1.44i·13-s + 16-s − 1.19i·19-s − 25-s + (0.188 − 0.981i)28-s + 1.86i·31-s − 1.80·37-s − 1.21·43-s + (−0.928 − 0.371i)49-s + 1.44i·52-s − 1.99i·61-s + 64-s + 0.610·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.981 + 0.188i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.981 + 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42294 - 0.135678i\)
\(L(\frac12)\) \(\approx\) \(1.42294 - 0.135678i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 2.59i)T \)
good2 \( 1 - 2T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 + 11T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 15.5iT - 61T^{2} \)
67 \( 1 - 5T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 15.5iT - 73T^{2} \)
79 \( 1 - 17T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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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

−12.34724376774533062031826258018, −11.44993374274485632181077985580, −10.76104049195539061702592615885, −9.715222368130451539338716214644, −8.404676935173241946472510450721, −7.08463020998111621518733980564, −6.66585914897251330746172488364, −4.96364125632316960128323566378, −3.53797810840314301150019977587, −1.79667931384199699777336608332, 2.05946601437150731894798750794, 3.41334002214837305559225227638, 5.43995460113139782364315789736, 6.14132080213471396691959727237, 7.60763725919989967678800386198, 8.338971354451228504286981770129, 9.806373410543754724936667305845, 10.66527287713730006787258232932, 11.77926194512005936178483948718, 12.31743923970937630724778633707

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