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Properties

Label 2-189-63.11-c2-0-1
Degree $2$
Conductor $189$
Sign $-0.889 + 0.457i$
Analytic cond. $5.14987$
Root an. cond. $2.26933$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.22i·2-s − 6.39·4-s + (4.79 + 2.76i)5-s + (−6.99 + 0.206i)7-s − 7.70i·8-s + (−8.91 + 15.4i)10-s + (−15.3 + 8.84i)11-s + (2.03 + 3.52i)13-s + (−0.665 − 22.5i)14-s − 0.715·16-s + (14.3 + 8.27i)17-s + (−3.92 − 6.79i)19-s + (−30.6 − 17.6i)20-s + (−28.5 − 49.3i)22-s + (−8.71 − 5.03i)23-s + ⋯
L(s)  = 1  + 1.61i·2-s − 1.59·4-s + (0.958 + 0.553i)5-s + (−0.999 + 0.0294i)7-s − 0.963i·8-s + (−0.891 + 1.54i)10-s + (−1.39 + 0.804i)11-s + (0.156 + 0.271i)13-s + (−0.0475 − 1.61i)14-s − 0.0447·16-s + (0.843 + 0.486i)17-s + (−0.206 − 0.357i)19-s + (−1.53 − 0.884i)20-s + (−1.29 − 2.24i)22-s + (−0.379 − 0.218i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.457i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.889 + 0.457i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ -0.889 + 0.457i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.270096 - 1.11654i\)
\(L(\frac12)\) \(\approx\) \(0.270096 - 1.11654i\)
\(L(2)\) 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 \( 1 + (6.99 - 0.206i)T \)
good2 \( 1 - 3.22iT - 4T^{2} \)
5 \( 1 + (-4.79 - 2.76i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (15.3 - 8.84i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-2.03 - 3.52i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-14.3 - 8.27i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (3.92 + 6.79i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (8.71 + 5.03i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-39.9 - 23.0i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 29.6T + 961T^{2} \)
37 \( 1 + (-15.5 - 27.0i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (27.8 - 16.0i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-3.35 + 5.80i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 16.4iT - 2.20e3T^{2} \)
53 \( 1 + (-32.5 - 18.8i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 - 95.0iT - 3.48e3T^{2} \)
61 \( 1 + 73.7T + 3.72e3T^{2} \)
67 \( 1 + 12.1T + 4.48e3T^{2} \)
71 \( 1 - 20.0iT - 5.04e3T^{2} \)
73 \( 1 + (11.4 - 19.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 138.T + 6.24e3T^{2} \)
83 \( 1 + (13.6 + 7.90i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-46.9 + 27.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-86.1 + 149. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.34379865027396690247637125176, −12.31179418180799247811216213926, −10.37934536092896063591004199591, −9.898728094363590772916200848935, −8.648125598785122823894669287679, −7.54708761542987138196497389337, −6.56145178802605088338692737448, −5.91460802082178767840107341513, −4.74220488853049606511335296591, −2.71099605686340535975479442824, 0.65931402932838934243427534678, 2.41015150500760987723104237708, 3.42390276993085494895470277946, 5.08114830313241261113727299495, 6.15562368140251867553964515271, 8.079878006444251111829495183221, 9.263526459121722997107228003196, 10.05981065547619523875369704685, 10.55304476082680344044216054788, 11.88726536518836856196834046382

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