Properties

Label 2-189-189.5-c1-0-14
Degree $2$
Conductor $189$
Sign $0.120 + 0.992i$
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  + (−2.44 + 0.431i)2-s + (1.58 − 0.703i)3-s + (3.93 − 1.43i)4-s + (0.403 − 2.28i)5-s + (−3.57 + 2.40i)6-s + (−0.412 − 2.61i)7-s + (−4.71 + 2.72i)8-s + (2.01 − 2.22i)9-s + 5.78i·10-s + (−5.86 + 1.03i)11-s + (5.22 − 5.03i)12-s + (−1.72 + 2.05i)13-s + (2.13 + 6.22i)14-s + (−0.971 − 3.90i)15-s + (3.96 − 3.32i)16-s + 1.48·17-s + ⋯
L(s)  = 1  + (−1.73 + 0.305i)2-s + (0.913 − 0.406i)3-s + (1.96 − 0.716i)4-s + (0.180 − 1.02i)5-s + (−1.45 + 0.982i)6-s + (−0.155 − 0.987i)7-s + (−1.66 + 0.962i)8-s + (0.670 − 0.742i)9-s + 1.82i·10-s + (−1.76 + 0.311i)11-s + (1.50 − 1.45i)12-s + (−0.478 + 0.570i)13-s + (0.571 + 1.66i)14-s + (−0.250 − 1.00i)15-s + (0.990 − 0.830i)16-s + 0.361·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.120 + 0.992i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.120 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.490906 - 0.434824i\)
\(L(\frac12)\) \(\approx\) \(0.490906 - 0.434824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.58 + 0.703i)T \)
7 \( 1 + (0.412 + 2.61i)T \)
good2 \( 1 + (2.44 - 0.431i)T + (1.87 - 0.684i)T^{2} \)
5 \( 1 + (-0.403 + 2.28i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (5.86 - 1.03i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (1.72 - 2.05i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 - 1.48T + 17T^{2} \)
19 \( 1 + 2.92iT - 19T^{2} \)
23 \( 1 + (-4.20 + 5.01i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-3.10 - 3.70i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.68 - 4.64i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (1.99 + 3.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.76 - 4.83i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-0.444 - 0.161i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-6.12 - 2.22i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-9.00 + 5.19i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.316 + 0.265i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.460 + 1.26i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.32 + 7.50i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-0.361 - 0.208i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.47 - 1.43i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.359 - 2.03i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (3.50 - 2.93i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + (0.857 - 2.35i)T + (-74.3 - 62.3i)T^{2} \)
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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

−12.48655622352116068954340014791, −10.79406197097848020375530307357, −10.05445466502692431307417613887, −9.136759955182488477617980476345, −8.390763847946434686336401849117, −7.51191741614273581478048222027, −6.81545618371330132501783839119, −4.82825640151141669998197573257, −2.54731270604411387154952185596, −0.923750825983386475196331056079, 2.44509959596348107621616615044, 2.95956800508096106069971922582, 5.59786643050718767190346101978, 7.29366942726785795781293066059, 7.960709520548472256940988448095, 8.838756995817174727049414622650, 9.971787126788479659471332080952, 10.31072661785493804547555192091, 11.28285556232761727813401175645, 12.59543055617374933746046448130

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