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Properties

Label 2-189-189.101-c1-0-15
Degree $2$
Conductor $189$
Sign $0.939 + 0.342i$
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  + (1.02 − 1.22i)2-s + (0.732 + 1.56i)3-s + (−0.0949 − 0.538i)4-s + (1.45 − 1.21i)5-s + (2.67 + 0.713i)6-s + (−2.50 − 0.864i)7-s + (2.00 + 1.15i)8-s + (−1.92 + 2.30i)9-s − 3.02i·10-s + (0.811 − 0.967i)11-s + (0.775 − 0.543i)12-s + (−0.326 + 0.898i)13-s + (−3.62 + 2.17i)14-s + (2.97 + 1.38i)15-s + (4.50 − 1.63i)16-s − 4.01·17-s + ⋯
L(s)  = 1  + (0.725 − 0.864i)2-s + (0.423 + 0.906i)3-s + (−0.0474 − 0.269i)4-s + (0.650 − 0.545i)5-s + (1.09 + 0.291i)6-s + (−0.945 − 0.326i)7-s + (0.710 + 0.410i)8-s + (−0.641 + 0.766i)9-s − 0.957i·10-s + (0.244 − 0.291i)11-s + (0.223 − 0.156i)12-s + (−0.0906 + 0.249i)13-s + (−0.967 + 0.580i)14-s + (0.769 + 0.358i)15-s + (1.12 − 0.409i)16-s − 0.973·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88314 - 0.332775i\)
\(L(\frac12)\) \(\approx\) \(1.88314 - 0.332775i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.732 - 1.56i)T \)
7 \( 1 + (2.50 + 0.864i)T \)
good2 \( 1 + (-1.02 + 1.22i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (-1.45 + 1.21i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-0.811 + 0.967i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.326 - 0.898i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 4.01T + 17T^{2} \)
19 \( 1 + 7.67iT - 19T^{2} \)
23 \( 1 + (1.95 - 5.37i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.22 + 3.37i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (8.76 - 1.54i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-3.99 + 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.01 - 1.09i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.111 + 0.632i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.48 - 8.42i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-11.3 - 6.54i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.02 + 0.738i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (3.03 + 0.535i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-1.63 + 1.37i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.154 - 0.0893i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-9.96 + 5.75i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.935 + 0.784i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-5.11 + 1.86i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 5.63T + 89T^{2} \)
97 \( 1 + (5.97 + 1.05i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.76940807577674753653860686869, −11.37328342546321417256791263292, −10.77826455471476216899731003377, −9.461009156508302739208479846227, −9.082054366487859103784749201530, −7.40666389421627444126654193995, −5.73144151194326799824544857928, −4.56462750201079273813425263325, −3.59141130298323071352191508553, −2.34509122160152829450190700226, 2.15875758703812398201031099387, 3.76757806383406135768087983332, 5.65712064028141611392136523097, 6.39968039532156376882712240979, 7.02891026725921017358032804747, 8.289344622942528470025321082906, 9.601925043166407129945645295088, 10.52267305365160087782645025766, 12.16728390301650304608364999868, 12.94172729968113663813403003816

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