Properties

Label 2-189-189.4-c1-0-16
Degree $2$
Conductor $189$
Sign $-0.704 + 0.709i$
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  + (−0.467 − 2.65i)2-s + (1.66 + 0.475i)3-s + (−4.92 + 1.79i)4-s + (2.75 − 1.00i)5-s + (0.482 − 4.63i)6-s + (−0.609 − 2.57i)7-s + (4.36 + 7.55i)8-s + (2.54 + 1.58i)9-s + (−3.94 − 6.82i)10-s + (−2.20 − 0.803i)11-s + (−9.05 + 0.642i)12-s + (−1.68 + 0.612i)13-s + (−6.53 + 2.81i)14-s + (5.06 − 0.358i)15-s + (9.95 − 8.35i)16-s + (−0.185 − 0.320i)17-s + ⋯
L(s)  = 1  + (−0.330 − 1.87i)2-s + (0.961 + 0.274i)3-s + (−2.46 + 0.896i)4-s + (1.23 − 0.448i)5-s + (0.197 − 1.89i)6-s + (−0.230 − 0.973i)7-s + (1.54 + 2.67i)8-s + (0.849 + 0.528i)9-s + (−1.24 − 2.15i)10-s + (−0.665 − 0.242i)11-s + (−2.61 + 0.185i)12-s + (−0.467 + 0.170i)13-s + (−1.74 + 0.753i)14-s + (1.30 − 0.0926i)15-s + (2.48 − 2.08i)16-s + (−0.0449 − 0.0777i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.704 + 0.709i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.704 + 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.505785 - 1.21548i\)
\(L(\frac12)\) \(\approx\) \(0.505785 - 1.21548i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.66 - 0.475i)T \)
7 \( 1 + (0.609 + 2.57i)T \)
good2 \( 1 + (0.467 + 2.65i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-2.75 + 1.00i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (2.20 + 0.803i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (1.68 - 0.612i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.185 + 0.320i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.496 - 0.860i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.14 - 6.50i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-6.76 - 2.46i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (3.45 - 1.25i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + 1.65T + 37T^{2} \)
41 \( 1 + (-8.90 + 3.23i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.36 + 7.73i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (0.234 + 0.0853i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (4.11 - 7.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.89 - 3.26i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.813 - 0.296i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.371 + 2.10i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-0.743 + 1.28i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 9.96T + 73T^{2} \)
79 \( 1 + (-1.60 - 9.09i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (6.30 + 2.29i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-6.39 + 11.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.55 - 14.4i)T + (-91.1 + 33.1i)T^{2} \)
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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

−12.33029190127295025215003429348, −10.82797863756018206332892728975, −10.17220741546008912055172923646, −9.543101515970748658186632422335, −8.766642231700112953766697973341, −7.56044350265370377479945996073, −5.18510043051683215742709943680, −3.97913449987185412334995248531, −2.76143402843893091726952686009, −1.53705434367240804379675764594, 2.51339400586238716758616627096, 4.77152524360010196345459018189, 6.00808952883012946177140919940, 6.67786639719703143247730969015, 7.86042329361206108981070223252, 8.691411298804779246512692760258, 9.592495526307767416694987506688, 10.17486296292158343007274911081, 12.68059203055685460673226375689, 13.27201984310572897165530910267

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