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Properties

Label 2-189-63.23-c2-0-3
Degree $2$
Conductor $189$
Sign $-0.921 + 0.389i$
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  + 2.74i·2-s − 3.53·4-s + (−2.32 + 1.34i)5-s + (0.122 + 6.99i)7-s + 1.27i·8-s + (−3.68 − 6.38i)10-s + (−7.18 − 4.14i)11-s + (1.91 − 3.31i)13-s + (−19.2 + 0.336i)14-s − 17.6·16-s + (−14.6 + 8.44i)17-s + (4.77 − 8.26i)19-s + (8.22 − 4.74i)20-s + (11.3 − 19.7i)22-s + (−21.1 + 12.2i)23-s + ⋯
L(s)  = 1  + 1.37i·2-s − 0.883·4-s + (−0.465 + 0.268i)5-s + (0.0174 + 0.999i)7-s + 0.159i·8-s + (−0.368 − 0.638i)10-s + (−0.653 − 0.377i)11-s + (0.147 − 0.255i)13-s + (−1.37 + 0.0240i)14-s − 1.10·16-s + (−0.860 + 0.496i)17-s + (0.251 − 0.435i)19-s + (0.411 − 0.237i)20-s + (0.517 − 0.896i)22-s + (−0.920 + 0.531i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.389i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.921 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.921 + 0.389i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ -0.921 + 0.389i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.201469 - 0.993649i\)
\(L(\frac12)\) \(\approx\) \(0.201469 - 0.993649i\)
\(L(2)\) 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 \( 1 + (-0.122 - 6.99i)T \)
good2 \( 1 - 2.74iT - 4T^{2} \)
5 \( 1 + (2.32 - 1.34i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (7.18 + 4.14i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-1.91 + 3.31i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (14.6 - 8.44i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-4.77 + 8.26i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (21.1 - 12.2i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-41.1 + 23.7i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 39.2T + 961T^{2} \)
37 \( 1 + (23.3 - 40.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-51.2 - 29.5i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-28.0 - 48.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 21.5iT - 2.20e3T^{2} \)
53 \( 1 + (22.2 - 12.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + 46.3iT - 3.48e3T^{2} \)
61 \( 1 - 60.3T + 3.72e3T^{2} \)
67 \( 1 + 64.0T + 4.48e3T^{2} \)
71 \( 1 - 49.6iT - 5.04e3T^{2} \)
73 \( 1 + (-58.7 - 101. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 - 40.5T + 6.24e3T^{2} \)
83 \( 1 + (-64.7 + 37.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-59.9 - 34.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (46.9 + 81.3i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.16065211329843506013462829345, −11.86162116447302620457563107254, −11.06434079468233160661681864830, −9.610506528300878664042677596532, −8.355965304047601354128284435846, −7.928763366553977796117418904800, −6.55886351855613767688036230208, −5.79871619436872452370894944663, −4.61031569930190864077087312638, −2.70421916487595063734459663256, 0.58022597172565553430457814960, 2.32224115867471215654766035145, 3.83719834583362973670600787690, 4.67284759918867538957493478234, 6.65273449861376555024942556423, 7.80542710641963006343409850804, 9.044825791407987647497575296957, 10.29680158354626668889442663641, 10.65017203285190254716614027435, 11.87906139336460429458417433498

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