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Properties

Label 2-189-189.101-c1-0-17
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: $\chi_{189} (101, \cdot )$
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

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

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