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Properties

Label 2-189-189.131-c1-0-7
Degree $2$
Conductor $189$
Sign $0.416 - 0.909i$
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  + (0.594 + 0.708i)2-s + (0.665 + 1.59i)3-s + (0.198 − 1.12i)4-s + (−0.386 − 0.324i)5-s + (−0.737 + 1.42i)6-s + (0.529 + 2.59i)7-s + (2.51 − 1.45i)8-s + (−2.11 + 2.12i)9-s − 0.466i·10-s + (3.12 + 3.72i)11-s + (1.93 − 0.432i)12-s + (−1.60 − 4.42i)13-s + (−1.52 + 1.91i)14-s + (0.261 − 0.833i)15-s + (0.374 + 0.136i)16-s − 4.77·17-s + ⋯
L(s)  = 1  + (0.420 + 0.500i)2-s + (0.384 + 0.923i)3-s + (0.0994 − 0.563i)4-s + (−0.172 − 0.144i)5-s + (−0.301 + 0.580i)6-s + (0.200 + 0.979i)7-s + (0.890 − 0.514i)8-s + (−0.705 + 0.709i)9-s − 0.147i·10-s + (0.941 + 1.12i)11-s + (0.558 − 0.124i)12-s + (−0.446 − 1.22i)13-s + (−0.406 + 0.512i)14-s + (0.0675 − 0.215i)15-s + (0.0936 + 0.0341i)16-s − 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.416 - 0.909i$
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: \((2,\ 189,\ (\ :1/2),\ 0.416 - 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38004 + 0.885581i\)
\(L(\frac12)\) \(\approx\) \(1.38004 + 0.885581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.665 - 1.59i)T \)
7 \( 1 + (-0.529 - 2.59i)T \)
good2 \( 1 + (-0.594 - 0.708i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (0.386 + 0.324i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (-3.12 - 3.72i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (1.60 + 4.42i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + 4.77T + 17T^{2} \)
19 \( 1 + 6.37iT - 19T^{2} \)
23 \( 1 + (-1.22 - 3.37i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.997 + 2.73i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (0.773 + 0.136i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (-2.73 - 4.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.44 + 1.98i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.36 + 7.75i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-1.33 - 7.56i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (4.26 - 2.46i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.01 - 1.82i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (7.07 - 1.24i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-6.70 - 5.62i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (11.3 + 6.53i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-8.42 - 4.86i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.63 - 3.05i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-2.75 - 1.00i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + (2.41 - 0.426i)T + (91.1 - 33.1i)T^{2} \)
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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

−12.91231013563441423130148362797, −11.69605279942541433376102008569, −10.70055252869656743475684580819, −9.634052241756684597754184680613, −8.952811977110692160501968363640, −7.59493310509100040223476684578, −6.27271682854950982630053135448, −5.06627579125993908079397041664, −4.38864518156869642807103188583, −2.45767576956587712689901178624, 1.73202041139082794387228596654, 3.36315672806447759836555538690, 4.29844881965644783304218801119, 6.38377980355121229688039602291, 7.24551899228589506493320854316, 8.206412135988377480499155917328, 9.205086307879135825551148695655, 10.96660933280486804333804189174, 11.49271371260223373425367241952, 12.42540822612976174498351623138

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