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Properties

Label 2-189-189.131-c1-0-5
Degree $2$
Conductor $189$
Sign $0.980 - 0.196i$
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  + (0.00959 + 0.0114i)2-s + (−1.04 + 1.38i)3-s + (0.347 − 1.96i)4-s + (1.26 + 1.05i)5-s + (−0.0258 + 0.00129i)6-s + (1.81 − 1.92i)7-s + (0.0517 − 0.0298i)8-s + (−0.814 − 2.88i)9-s + 0.0245i·10-s + (2.95 + 3.51i)11-s + (2.35 + 2.53i)12-s + (1.52 + 4.19i)13-s + (0.0394 + 0.00230i)14-s + (−2.77 + 0.635i)15-s + (−3.75 − 1.36i)16-s + 3.77·17-s + ⋯
L(s)  = 1  + (0.00678 + 0.00808i)2-s + (−0.603 + 0.797i)3-s + (0.173 − 0.984i)4-s + (0.563 + 0.473i)5-s + (−0.0105 + 0.000528i)6-s + (0.686 − 0.727i)7-s + (0.0182 − 0.0105i)8-s + (−0.271 − 0.962i)9-s + 0.00776i·10-s + (0.890 + 1.06i)11-s + (0.680 + 0.732i)12-s + (0.423 + 1.16i)13-s + (0.0105 + 0.000616i)14-s + (−0.717 + 0.163i)15-s + (−0.939 − 0.341i)16-s + 0.915·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.980 - 0.196i$
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: \((2,\ 189,\ (\ :1/2),\ 0.980 - 0.196i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19367 + 0.118523i\)
\(L(\frac12)\) \(\approx\) \(1.19367 + 0.118523i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.04 - 1.38i)T \)
7 \( 1 + (-1.81 + 1.92i)T \)
good2 \( 1 + (-0.00959 - 0.0114i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (-1.26 - 1.05i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (-2.95 - 3.51i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-1.52 - 4.19i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 - 3.77T + 17T^{2} \)
19 \( 1 + 1.46iT - 19T^{2} \)
23 \( 1 + (2.54 + 6.98i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (2.83 - 7.78i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (8.27 + 1.45i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (-0.397 - 0.688i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.97 - 1.44i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.303 + 1.71i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (0.286 + 1.62i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (3.66 - 2.11i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.8 - 3.95i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-2.96 + 0.522i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (8.30 + 6.96i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-6.66 - 3.84i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-13.7 - 7.91i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.22 + 1.02i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (4.61 + 1.68i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + 7.18T + 89T^{2} \)
97 \( 1 + (-1.56 + 0.276i)T + (91.1 - 33.1i)T^{2} \)
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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

−12.33029171524768363878329649840, −11.29562315442919268404939466922, −10.60283263435900059498547675037, −9.858697120668681063073382457485, −9.027353313161516459586703988045, −7.01875013086572439078374518408, −6.31370019616711021508512460661, −5.04082481497808894311286387731, −4.08011699255654412420452870170, −1.66914650186055983857923072327, 1.66200606636842504355266634588, 3.44339928074997859590381428925, 5.42833516579746877713901206672, 6.02185659635284394707350406406, 7.63864573639360881733791160591, 8.232516608585780814236877561465, 9.337778293316655379934010434783, 11.03853723228283597070312234596, 11.68192749320483318357210680258, 12.47346835353423082201774036653

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