Properties

Label 2-189-189.5-c1-0-2
Degree $2$
Conductor $189$
Sign $-0.961 - 0.275i$
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  + (−1.19 + 0.211i)2-s + (−0.0787 + 1.73i)3-s + (−0.485 + 0.176i)4-s + (−0.243 + 1.37i)5-s + (−0.271 − 2.09i)6-s + (2.41 + 1.08i)7-s + (2.65 − 1.53i)8-s + (−2.98 − 0.272i)9-s − 1.70i·10-s + (−6.43 + 1.13i)11-s + (−0.267 − 0.853i)12-s + (−0.697 + 0.831i)13-s + (−3.12 − 0.792i)14-s + (−2.36 − 0.529i)15-s + (−2.06 + 1.73i)16-s − 1.26·17-s + ⋯
L(s)  = 1  + (−0.848 + 0.149i)2-s + (−0.0454 + 0.998i)3-s + (−0.242 + 0.0882i)4-s + (−0.108 + 0.616i)5-s + (−0.110 − 0.854i)6-s + (0.911 + 0.410i)7-s + (0.938 − 0.541i)8-s + (−0.995 − 0.0908i)9-s − 0.539i·10-s + (−1.93 + 0.341i)11-s + (−0.0771 − 0.246i)12-s + (−0.193 + 0.230i)13-s + (−0.834 − 0.211i)14-s + (−0.611 − 0.136i)15-s + (−0.517 + 0.434i)16-s − 0.305·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.961 - 0.275i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ -0.961 - 0.275i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0697652 + 0.496642i\)
\(L(\frac12)\) \(\approx\) \(0.0697652 + 0.496642i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0787 - 1.73i)T \)
7 \( 1 + (-2.41 - 1.08i)T \)
good2 \( 1 + (1.19 - 0.211i)T + (1.87 - 0.684i)T^{2} \)
5 \( 1 + (0.243 - 1.37i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (6.43 - 1.13i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (0.697 - 0.831i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + 1.26T + 17T^{2} \)
19 \( 1 + 3.25iT - 19T^{2} \)
23 \( 1 + (2.13 - 2.54i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-1.95 - 2.33i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.14 - 3.13i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (-2.93 - 5.08i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.04 - 0.875i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (4.70 + 1.71i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-8.38 - 3.05i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-7.98 + 4.61i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.516 - 0.433i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (2.39 - 6.56i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.92 - 10.9i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-1.73 - 0.999i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (12.1 + 6.98i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.78 - 10.1i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (7.80 - 6.54i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 + (-6.09 + 16.7i)T + (-74.3 - 62.3i)T^{2} \)
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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

−13.09398922572357971248848290864, −11.64369884778143099356511733105, −10.64144002496925776455159231897, −10.17503983032508585893176795513, −8.975834667154775861910627407774, −8.195521233890197474263186025166, −7.22431182844451145876589161496, −5.33083641588804486273396879136, −4.50363213590350489502691305528, −2.72701992759044745564278627299, 0.59313223609174539146710944629, 2.23154689169751699522904033690, 4.68873654334940046383009719581, 5.67447662580093815189748248347, 7.56178467649550426342876396742, 8.048805125207841792077930640712, 8.762760570869906632711090439029, 10.29397853197134515332026329370, 10.91873966478200417285893353275, 12.13861521858451553182283069096

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