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Properties

Label 2-189-63.38-c1-0-1
Degree $2$
Conductor $189$
Sign $0.383 - 0.923i$
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  + 0.293i·2-s + 1.91·4-s + (−1.53 + 2.65i)5-s + (−1.41 + 2.23i)7-s + 1.15i·8-s + (−0.778 − 0.449i)10-s + (3.37 − 1.94i)11-s + (−2.02 + 1.17i)13-s + (−0.656 − 0.416i)14-s + 3.48·16-s + (1.68 − 2.91i)17-s + (2.20 − 1.27i)19-s + (−2.92 + 5.07i)20-s + (0.572 + 0.991i)22-s + (−2.58 − 1.49i)23-s + ⋯
L(s)  = 1  + 0.207i·2-s + 0.956·4-s + (−0.684 + 1.18i)5-s + (−0.535 + 0.844i)7-s + 0.406i·8-s + (−0.246 − 0.142i)10-s + (1.01 − 0.587i)11-s + (−0.562 + 0.324i)13-s + (−0.175 − 0.111i)14-s + 0.872·16-s + (0.408 − 0.706i)17-s + (0.506 − 0.292i)19-s + (−0.654 + 1.13i)20-s + (0.122 + 0.211i)22-s + (−0.538 − 0.310i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.383 - 0.923i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.383 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04756 + 0.699034i\)
\(L(\frac12)\) \(\approx\) \(1.04756 + 0.699034i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - 2.23i)T \)
good2 \( 1 - 0.293iT - 2T^{2} \)
5 \( 1 + (1.53 - 2.65i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.37 + 1.94i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.02 - 1.17i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.68 + 2.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.20 + 1.27i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.58 + 1.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.67 - 2.12i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.472iT - 31T^{2} \)
37 \( 1 + (3.89 + 6.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.12 + 5.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.06 + 3.57i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.05T + 47T^{2} \)
53 \( 1 + (4.99 + 2.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.68T + 59T^{2} \)
61 \( 1 - 1.60iT - 61T^{2} \)
67 \( 1 - 1.57T + 67T^{2} \)
71 \( 1 - 13.6iT - 71T^{2} \)
73 \( 1 + (0.856 + 0.494i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 9.27T + 79T^{2} \)
83 \( 1 + (5.49 - 9.51i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.15 - 3.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.98 - 2.87i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.29790729852557125865580946147, −11.77078264371106290337282404331, −11.01163554347060992457218832410, −9.889623183639662692761473107170, −8.646562688651250649073667617536, −7.27348954256545652866850352877, −6.73666696534749416202096504867, −5.63720660146986862478004104941, −3.55483215718837896884604954022, −2.54382483723487821957040598312, 1.30861482930638316345293357177, 3.45372605626832208034267014776, 4.57223048296032962226624639629, 6.19935175770181282556358286705, 7.30966271896098527781196743908, 8.172871363920180717075324122665, 9.600427208183955920127934811020, 10.35872964179586984140170605708, 11.72685309040547911333876841956, 12.20969586084341950425391593067

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