Properties

Label 2-189-189.184-c1-0-10
Degree $2$
Conductor $189$
Sign $-0.548 + 0.835i$
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  + (−0.303 − 1.72i)2-s + (−1.25 + 1.19i)3-s + (−0.999 + 0.363i)4-s + (0.0231 − 0.131i)5-s + (2.44 + 1.79i)6-s + (1.74 − 1.99i)7-s + (−0.819 − 1.41i)8-s + (0.140 − 2.99i)9-s − 0.233·10-s + (−0.550 − 3.11i)11-s + (0.817 − 1.65i)12-s + (−1.13 − 0.948i)13-s + (−3.96 − 2.40i)14-s + (0.128 + 0.192i)15-s + (−3.82 + 3.21i)16-s − 0.883·17-s + ⋯
L(s)  = 1  + (−0.214 − 1.21i)2-s + (−0.723 + 0.690i)3-s + (−0.499 + 0.181i)4-s + (0.0103 − 0.0588i)5-s + (0.996 + 0.733i)6-s + (0.658 − 0.752i)7-s + (−0.289 − 0.501i)8-s + (0.0469 − 0.998i)9-s − 0.0739·10-s + (−0.165 − 0.940i)11-s + (0.235 − 0.476i)12-s + (−0.313 − 0.263i)13-s + (−1.05 − 0.641i)14-s + (0.0331 + 0.0497i)15-s + (−0.956 + 0.802i)16-s − 0.214·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 + 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.548 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.548 + 0.835i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.548 + 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.405655 - 0.751756i\)
\(L(\frac12)\) \(\approx\) \(0.405655 - 0.751756i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.25 - 1.19i)T \)
7 \( 1 + (-1.74 + 1.99i)T \)
good2 \( 1 + (0.303 + 1.72i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-0.0231 + 0.131i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (0.550 + 3.11i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (1.13 + 0.948i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + 0.883T + 17T^{2} \)
19 \( 1 - 6.03T + 19T^{2} \)
23 \( 1 + (3.25 + 2.72i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-2.60 + 2.18i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (5.70 - 2.07i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-1.87 - 3.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.09 - 5.11i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (6.38 + 2.32i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (3.12 + 1.13i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-6.45 - 11.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.32 - 4.47i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-12.1 - 4.40i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.662 + 3.75i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (2.23 - 3.86i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.0471 + 0.0817i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.562 + 3.19i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (8.18 - 6.86i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + 3.76T + 89T^{2} \)
97 \( 1 + (14.4 + 5.25i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.78396083917421070851276357665, −11.21960626976301032190196867617, −10.45777233114890678144671481091, −9.744146158770769081875580483347, −8.549448988183856731429430135390, −7.00079988653280464210228962183, −5.58498420785581460636375359786, −4.31553451966141189873561220616, −3.11622963313429279071286343872, −0.945952340514523848975573898561, 2.17793339843182798014152131592, 4.95346137282388869763720752603, 5.63368347098712257242012586948, 6.86229452773178294441254017597, 7.52955641918579276449085726495, 8.477406238251151155648160923437, 9.727031518885579173199142155856, 11.24496036861459497567605362896, 11.89738416509705480505779198511, 12.82552862382414824717546554144

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