Properties

Label 2-189-27.22-c1-0-9
Degree $2$
Conductor $189$
Sign $0.892 - 0.451i$
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  + (0.0808 − 0.0678i)2-s + (1.71 − 0.236i)3-s + (−0.345 + 1.95i)4-s + (0.0449 − 0.0163i)5-s + (0.122 − 0.135i)6-s + (0.173 + 0.984i)7-s + (0.210 + 0.364i)8-s + (2.88 − 0.811i)9-s + (0.00252 − 0.00437i)10-s + (−2.08 − 0.757i)11-s + (−0.129 + 3.44i)12-s + (1.63 + 1.36i)13-s + (0.0808 + 0.0678i)14-s + (0.0732 − 0.0386i)15-s + (−3.69 − 1.34i)16-s + (1.30 − 2.25i)17-s + ⋯
L(s)  = 1  + (0.0571 − 0.0479i)2-s + (0.990 − 0.136i)3-s + (−0.172 + 0.979i)4-s + (0.0200 − 0.00731i)5-s + (0.0501 − 0.0553i)6-s + (0.0656 + 0.372i)7-s + (0.0744 + 0.128i)8-s + (0.962 − 0.270i)9-s + (0.000798 − 0.00138i)10-s + (−0.627 − 0.228i)11-s + (−0.0374 + 0.993i)12-s + (0.452 + 0.379i)13-s + (0.0216 + 0.0181i)14-s + (0.0189 − 0.00998i)15-s + (−0.924 − 0.336i)16-s + (0.315 − 0.547i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.892 - 0.451i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.892 - 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51486 + 0.361762i\)
\(L(\frac12)\) \(\approx\) \(1.51486 + 0.361762i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 + 0.236i)T \)
7 \( 1 + (-0.173 - 0.984i)T \)
good2 \( 1 + (-0.0808 + 0.0678i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (-0.0449 + 0.0163i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (2.08 + 0.757i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-1.63 - 1.36i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.30 + 2.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.12 + 1.95i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.36 + 7.75i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-0.516 + 0.433i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.714 - 4.05i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-1.50 + 2.60i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.51 + 3.78i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (7.25 + 2.63i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.831 - 4.71i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 2.27T + 53T^{2} \)
59 \( 1 + (11.3 - 4.11i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.115 - 0.657i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-7.86 - 6.59i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-7.10 + 12.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.97 - 3.42i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.63 + 2.21i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (9.22 - 7.74i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-5.98 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-17.1 - 6.24i)T + (74.3 + 62.3i)T^{2} \)
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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

−12.77499954450228424258961893323, −11.92145581264318843173496206646, −10.68002611738487981815144115679, −9.304443771245625688691630329989, −8.583405030092438069486883874347, −7.76312102964244078706732579700, −6.69006944080885798322487807879, −4.82930586938468594141845891563, −3.52204352308532633763557169012, −2.41565934092498334286484954983, 1.74023174146940542564654360416, 3.55105036511507037060132628840, 4.86799517597914330681881120315, 6.14175576764400421005339955868, 7.54983199192099480053509428455, 8.448287046534552199319682238849, 9.749184119022110274113568081294, 10.17923008185316578433358279293, 11.30460725613912831229504888180, 12.90257224028133507890663142492

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