Properties

Label 2-189-9.4-c1-0-0
Degree $2$
Conductor $189$
Sign $0.991 - 0.128i$
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  + (−1.23 − 2.13i)2-s + (−2.02 + 3.51i)4-s + (−1.29 + 2.24i)5-s + (0.5 + 0.866i)7-s + 5.05·8-s + 6.38·10-s + (2.25 + 3.90i)11-s + (−0.5 + 0.866i)13-s + (1.23 − 2.13i)14-s + (−2.16 − 3.74i)16-s + 0.945·17-s − 4.05·19-s + (−5.25 − 9.10i)20-s + (5.55 − 9.61i)22-s + (−0.136 + 0.236i)23-s + ⋯
L(s)  = 1  + (−0.869 − 1.50i)2-s + (−1.01 + 1.75i)4-s + (−0.579 + 1.00i)5-s + (0.188 + 0.327i)7-s + 1.78·8-s + 2.01·10-s + (0.680 + 1.17i)11-s + (−0.138 + 0.240i)13-s + (0.328 − 0.569i)14-s + (−0.540 − 0.936i)16-s + 0.229·17-s − 0.930·19-s + (−1.17 − 2.03i)20-s + (1.18 − 2.05i)22-s + (−0.0284 + 0.0493i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.991 - 0.128i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.991 - 0.128i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.579132 + 0.0373941i\)
\(L(\frac12)\) \(\approx\) \(0.579132 + 0.0373941i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (1.23 + 2.13i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.29 - 2.24i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.25 - 3.90i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.945T + 17T^{2} \)
19 \( 1 + 4.05T + 19T^{2} \)
23 \( 1 + (0.136 - 0.236i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.23 - 2.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.16 - 2.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.78T + 37T^{2} \)
41 \( 1 + (3.20 - 5.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.21 - 9.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.08 + 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.27T + 53T^{2} \)
59 \( 1 + (1.36 - 2.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.13 - 1.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.90 + 13.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.27T + 71T^{2} \)
73 \( 1 + 1.50T + 73T^{2} \)
79 \( 1 + (7.35 + 12.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.472 + 0.819i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + (-5.74 - 9.95i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.13648112448905929139164286265, −11.52457526734605264945426642266, −10.66210608846542098329298094021, −9.848614979075890418744003417622, −8.891878641754838243450151713282, −7.77756478730675404803854954086, −6.67953172013841081901892474199, −4.42872773456760787609298956683, −3.22831514581994555807523278061, −1.92785582544017823996418248887, 0.72305490071213197993133300463, 4.12623441675620218959095665002, 5.40355588632900731165398578474, 6.43407349601293785210187713794, 7.63015493574724958754094281830, 8.473742459181020037324169462356, 8.987342120003008759873443637076, 10.23296004755962489788738481906, 11.45708913073124502125238040765, 12.67546146885602992690220911711

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