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Properties

Label 2-189-63.58-c1-0-5
Degree $2$
Conductor $189$
Sign $-0.245 + 0.969i$
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  + 0.495·2-s − 1.75·4-s + (−1.84 − 3.19i)5-s + (0.926 − 2.47i)7-s − 1.86·8-s + (−0.915 − 1.58i)10-s + (−0.446 + 0.772i)11-s + (0.598 − 1.03i)13-s + (0.459 − 1.22i)14-s + 2.58·16-s + (0.124 + 0.216i)17-s + (1.40 − 2.43i)19-s + (3.23 + 5.60i)20-s + (−0.221 + 0.383i)22-s + (1.23 + 2.14i)23-s + ⋯
L(s)  = 1  + 0.350·2-s − 0.877·4-s + (−0.825 − 1.43i)5-s + (0.350 − 0.936i)7-s − 0.658·8-s + (−0.289 − 0.501i)10-s + (−0.134 + 0.233i)11-s + (0.165 − 0.287i)13-s + (0.122 − 0.328i)14-s + 0.646·16-s + (0.0303 + 0.0525i)17-s + (0.322 − 0.557i)19-s + (0.724 + 1.25i)20-s + (−0.0471 + 0.0817i)22-s + (0.258 + 0.447i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.245 + 0.969i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.245 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.548608 - 0.704861i\)
\(L(\frac12)\) \(\approx\) \(0.548608 - 0.704861i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.926 + 2.47i)T \)
good2 \( 1 - 0.495T + 2T^{2} \)
5 \( 1 + (1.84 + 3.19i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.446 - 0.772i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.598 + 1.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.124 - 0.216i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.40 + 2.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.23 - 2.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.07 + 3.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.58T + 31T^{2} \)
37 \( 1 + (2.36 - 4.09i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.39 + 4.14i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.98 + 8.64i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 + (-4.94 - 8.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 1.81T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 - 1.02T + 67T^{2} \)
71 \( 1 - 4.94T + 71T^{2} \)
73 \( 1 + (0.915 + 1.58i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 1.79T + 79T^{2} \)
83 \( 1 + (6.16 + 10.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.20 + 2.08i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.52 - 9.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.41205063677680396238732468891, −11.60085693060573983290134399058, −10.24229593967506130801910991769, −9.097012080537321441418444376299, −8.311983876201079429917226847678, −7.37614100374515767314471176284, −5.44467831442530836385748398724, −4.57851465913554016565264306493, −3.76545768748717158411161479561, −0.77591442529177466918848278731, 2.85252598526421973104828269099, 3.96019931308712359055381105532, 5.35760663634513773950964917575, 6.54039099138433021193336481500, 7.84107493133968333407964401189, 8.735234096776101178125199718269, 9.941422668568974061768634713241, 11.08101209411915664549597554708, 11.84767788559973961303714966901, 12.79265142101432031800126763379

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