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Properties

Label 2-189-63.38-c1-0-4
Degree $2$
Conductor $189$
Sign $-0.370 + 0.928i$
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  − 1.51i·2-s − 0.280·4-s + (0.387 − 0.671i)5-s + (−2.46 − 0.952i)7-s − 2.59i·8-s + (−1.01 − 0.585i)10-s + (3.32 − 1.92i)11-s + (2.54 − 1.46i)13-s + (−1.43 + 3.72i)14-s − 4.48·16-s + (−2.69 + 4.67i)17-s + (−0.376 + 0.217i)19-s + (−0.108 + 0.188i)20-s + (−2.90 − 5.02i)22-s + (0.0482 + 0.0278i)23-s + ⋯
L(s)  = 1  − 1.06i·2-s − 0.140·4-s + (0.173 − 0.300i)5-s + (−0.933 − 0.359i)7-s − 0.918i·8-s + (−0.320 − 0.185i)10-s + (1.00 − 0.579i)11-s + (0.705 − 0.407i)13-s + (−0.384 + 0.996i)14-s − 1.12·16-s + (−0.654 + 1.13i)17-s + (−0.0863 + 0.0498i)19-s + (−0.0243 + 0.0421i)20-s + (−0.618 − 1.07i)22-s + (0.0100 + 0.00580i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.370 + 0.928i$
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: \((2,\ 189,\ (\ :1/2),\ -0.370 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.696144 - 1.02727i\)
\(L(\frac12)\) \(\approx\) \(0.696144 - 1.02727i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.46 + 0.952i)T \)
good2 \( 1 + 1.51iT - 2T^{2} \)
5 \( 1 + (-0.387 + 0.671i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.32 + 1.92i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.54 + 1.46i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.69 - 4.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.376 - 0.217i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0482 - 0.0278i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.187 - 0.108i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.55iT - 31T^{2} \)
37 \( 1 + (-3.14 - 5.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.78 - 6.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.42 + 11.1i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.965T + 47T^{2} \)
53 \( 1 + (6.46 + 3.73i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 3.12T + 59T^{2} \)
61 \( 1 - 3.48iT - 61T^{2} \)
67 \( 1 + 4.20T + 67T^{2} \)
71 \( 1 + 3.50iT - 71T^{2} \)
73 \( 1 + (7.05 + 4.07i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 4.96T + 79T^{2} \)
83 \( 1 + (4.31 - 7.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.82 - 13.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.24 + 0.716i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.29982684324114834416664094233, −11.16352083650379991078246020829, −10.51721956867404760954171195186, −9.472797458511662111868803578579, −8.593417969379287285844883850343, −6.87474972607109651128057942725, −6.04690544116509703391573017521, −4.07519694612709917997536643997, −3.15213173408645171662897643194, −1.29802479762276862769929722663, 2.53051663551318218978480551630, 4.34756043822478548631713946859, 5.95434805869362706075894880998, 6.57530251953077841440941819800, 7.44897803607204274982199335869, 8.880557476700371696821109971876, 9.540957218860457147474985522436, 11.00743084105773203937261114113, 11.87050602513246465682388762881, 13.04240781492581768406904129472

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