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Properties

Label 2-189-189.131-c1-0-10
Degree $2$
Conductor $189$
Sign $0.925 - 0.379i$
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  + (−0.0103 − 0.0123i)2-s + (1.25 + 1.19i)3-s + (0.347 − 1.96i)4-s + (3.05 + 2.56i)5-s + (0.00168 − 0.0279i)6-s + (−2.57 − 0.614i)7-s + (−0.0559 + 0.0323i)8-s + (0.163 + 2.99i)9-s − 0.0645i·10-s + (−3.31 − 3.95i)11-s + (2.78 − 2.06i)12-s + (0.220 + 0.606i)13-s + (0.0191 + 0.0382i)14-s + (0.789 + 6.86i)15-s + (−3.75 − 1.36i)16-s + 3.38·17-s + ⋯
L(s)  = 1  + (−0.00734 − 0.00875i)2-s + (0.726 + 0.687i)3-s + (0.173 − 0.984i)4-s + (1.36 + 1.14i)5-s + (0.000686 − 0.0114i)6-s + (−0.972 − 0.232i)7-s + (−0.0197 + 0.0114i)8-s + (0.0543 + 0.998i)9-s − 0.0204i·10-s + (−0.999 − 1.19i)11-s + (0.803 − 0.595i)12-s + (0.0612 + 0.168i)13-s + (0.00511 + 0.0102i)14-s + (0.203 + 1.77i)15-s + (−0.939 − 0.341i)16-s + 0.821·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.925 - 0.379i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.925 - 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56266 + 0.308299i\)
\(L(\frac12)\) \(\approx\) \(1.56266 + 0.308299i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.25 - 1.19i)T \)
7 \( 1 + (2.57 + 0.614i)T \)
good2 \( 1 + (0.0103 + 0.0123i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (-3.05 - 2.56i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (3.31 + 3.95i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-0.220 - 0.606i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 - 3.38T + 17T^{2} \)
19 \( 1 - 0.379iT - 19T^{2} \)
23 \( 1 + (-0.344 - 0.947i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-1.97 + 5.41i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (6.35 + 1.12i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (3.08 + 5.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.266 - 0.0969i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.45 + 8.22i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-1.27 - 7.24i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (1.71 - 0.992i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.21 - 1.17i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (7.93 - 1.39i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-7.23 - 6.06i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-8.57 - 4.94i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.13 + 3.54i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.980 - 0.823i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-8.07 - 2.93i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + 5.35T + 89T^{2} \)
97 \( 1 + (-7.39 + 1.30i)T + (91.1 - 33.1i)T^{2} \)
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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

−13.12938075361344828359523243608, −11.03185521069504724429809505998, −10.43375450120324849442280171257, −9.865696008268747507739352133224, −9.058861274395361412612748265846, −7.38078146590563202178656243120, −6.09368044570191010266652227516, −5.47841335584981827565398841302, −3.35997572789390557936019573130, −2.34061196717170909444253951215, 1.97158095188852955189076121030, 3.16586438578783213824571773672, 5.02139049173617986736981113553, 6.38577768236504876773902537857, 7.46890149694342437781583437472, 8.542616361318278012178399791404, 9.348078854894070368328195431306, 10.14876689139220289146237966169, 12.22421695342073369197684024353, 12.73550312687747448706395228531

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