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Properties

Label 2-189-63.20-c3-0-12
Degree $2$
Conductor $189$
Sign $-0.952 + 0.306i$
Analytic cond. $11.1513$
Root an. cond. $3.33936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.27 − 2.46i)2-s + (8.16 + 14.1i)4-s + (−7.65 − 13.2i)5-s + (11.9 − 14.1i)7-s − 41.0i·8-s + 75.5i·10-s + (36.9 + 21.3i)11-s + (24.2 − 14.0i)13-s + (−85.8 + 31.2i)14-s + (−35.9 + 62.2i)16-s + 82.0·17-s − 113. i·19-s + (125. − 216. i)20-s + (−105. − 182. i)22-s + (25.2 − 14.5i)23-s + ⋯
L(s)  = 1  + (−1.51 − 0.871i)2-s + (1.02 + 1.76i)4-s + (−0.684 − 1.18i)5-s + (0.643 − 0.765i)7-s − 1.81i·8-s + 2.38i·10-s + (1.01 + 0.584i)11-s + (0.517 − 0.299i)13-s + (−1.63 + 0.595i)14-s + (−0.562 + 0.973i)16-s + 1.17·17-s − 1.36i·19-s + (1.39 − 2.42i)20-s + (−1.01 − 1.76i)22-s + (0.228 − 0.132i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.306i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.952 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.952 + 0.306i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ -0.952 + 0.306i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.112929 - 0.720278i\)
\(L(\frac12)\) \(\approx\) \(0.112929 - 0.720278i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11.9 + 14.1i)T \)
good2 \( 1 + (4.27 + 2.46i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (7.65 + 13.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-36.9 - 21.3i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-24.2 + 14.0i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 82.0T + 4.91e3T^{2} \)
19 \( 1 + 113. iT - 6.85e3T^{2} \)
23 \( 1 + (-25.2 + 14.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (16.5 + 9.56i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (97.0 - 56.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 69.2T + 5.06e4T^{2} \)
41 \( 1 + (242. + 419. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (194. - 337. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (53.4 - 92.5i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 207. iT - 1.48e5T^{2} \)
59 \( 1 + (97.8 + 169. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (446. + 257. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-221. - 383. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 640. iT - 3.57e5T^{2} \)
73 \( 1 + 528. iT - 3.89e5T^{2} \)
79 \( 1 + (-20.7 + 35.8i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-554. + 960. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 121.T + 7.04e5T^{2} \)
97 \( 1 + (-748. - 432. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.54221050346930993613271253198, −10.66368824948109823927234316781, −9.558650050923374687698668953976, −8.755846021923958190278852183809, −7.973505794124667097157508558473, −7.07326917668913532302740985731, −4.83929061401733582190805776721, −3.57927347441681981414770054560, −1.51390498395823660341409209299, −0.61917132801838653574483851636, 1.48151577523860861258065426970, 3.50699553668340741738073009109, 5.74613838795986229510400115906, 6.59040743043884325396611208067, 7.67719888050207534107179334350, 8.342671433699631608014117807142, 9.354272530423735634672215188079, 10.42468264494619229278660533536, 11.30190159326159511799038898343, 12.00809998874193603674352173885

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