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Properties

Label 2-189-63.61-c2-0-0
Degree $2$
Conductor $189$
Sign $-0.182 - 0.983i$
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  + (−0.662 − 1.14i)2-s + (1.12 − 1.94i)4-s + 7.23i·5-s + (−3.90 + 5.80i)7-s − 8.27·8-s + (8.30 − 4.79i)10-s − 17.1·11-s + (−9.05 + 5.22i)13-s + (9.25 + 0.637i)14-s + (0.990 + 1.71i)16-s + (−5.40 + 3.11i)17-s + (10.2 + 5.93i)19-s + (14.0 + 8.12i)20-s + (11.3 + 19.6i)22-s + 11.7·23-s + ⋯
L(s)  = 1  + (−0.331 − 0.573i)2-s + (0.280 − 0.486i)4-s + 1.44i·5-s + (−0.558 + 0.829i)7-s − 1.03·8-s + (0.830 − 0.479i)10-s − 1.55·11-s + (−0.696 + 0.402i)13-s + (0.660 + 0.0455i)14-s + (0.0618 + 0.107i)16-s + (−0.317 + 0.183i)17-s + (0.541 + 0.312i)19-s + (0.703 + 0.406i)20-s + (0.515 + 0.893i)22-s + 0.509·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.182 - 0.983i$
Analytic conductor: \(5.14987\)
Root analytic conductor: \(2.26933\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1),\ -0.182 - 0.983i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.396016 + 0.476243i\)
\(L(\frac12)\) \(\approx\) \(0.396016 + 0.476243i\)
\(L(2)\) 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 \( 1 + (3.90 - 5.80i)T \)
good2 \( 1 + (0.662 + 1.14i)T + (-2 + 3.46i)T^{2} \)
5 \( 1 - 7.23iT - 25T^{2} \)
11 \( 1 + 17.1T + 121T^{2} \)
13 \( 1 + (9.05 - 5.22i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (5.40 - 3.11i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-10.2 - 5.93i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 - 11.7T + 529T^{2} \)
29 \( 1 + (-5.48 + 9.50i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-24.3 - 14.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (33.7 - 58.4i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-14.1 + 8.17i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-31.2 + 54.2i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-26.1 + 15.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-11.3 - 19.6i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (1.58 + 0.916i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (12.9 - 7.49i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (29.7 - 51.4i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 6.84T + 5.04e3T^{2} \)
73 \( 1 + (75.3 - 43.5i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (34.3 + 59.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (30.2 + 17.4i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-12.9 - 7.45i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (41.2 + 23.8i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−12.27969968382650459839820725552, −11.50690523334893150015063115035, −10.37481915347796249920613231158, −10.13743802423742214905486213839, −8.845372376629828782899933895800, −7.34763844918795725551305503467, −6.39174163283967795039955303845, −5.33356624629776974123538924966, −3.01894501029537975673475065019, −2.39238688727403123701387153544, 0.36848924432570857375206875699, 2.87566694955541611812338429975, 4.57178251787507046843339139448, 5.68183178329414564732061294331, 7.21374146645858470388870401169, 7.86652695112119302353644245057, 8.886189772686494004040469643384, 9.834386941148207197820704020181, 11.05343027686055925260002645694, 12.47506912389625186893175633207

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