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Properties

Label 2-189-63.38-c1-0-3
Degree $2$
Conductor $189$
Sign $0.969 - 0.246i$
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  + 0.718i·2-s + 1.48·4-s + (0.723 − 1.25i)5-s + (0.182 − 2.63i)7-s + 2.50i·8-s + (0.900 + 0.519i)10-s + (1.55 − 0.900i)11-s + (−1.88 + 1.09i)13-s + (1.89 + 0.131i)14-s + 1.17·16-s + (−1.95 + 3.38i)17-s + (−3.47 + 2.00i)19-s + (1.07 − 1.86i)20-s + (0.646 + 1.11i)22-s + (4.91 + 2.83i)23-s + ⋯
L(s)  = 1  + 0.507i·2-s + 0.742·4-s + (0.323 − 0.560i)5-s + (0.0690 − 0.997i)7-s + 0.884i·8-s + (0.284 + 0.164i)10-s + (0.470 − 0.271i)11-s + (−0.523 + 0.302i)13-s + (0.506 + 0.0350i)14-s + 0.292·16-s + (−0.473 + 0.820i)17-s + (−0.797 + 0.460i)19-s + (0.240 − 0.416i)20-s + (0.137 + 0.238i)22-s + (1.02 + 0.591i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.969 - 0.246i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.969 - 0.246i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42609 + 0.178682i\)
\(L(\frac12)\) \(\approx\) \(1.42609 + 0.178682i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.182 + 2.63i)T \)
good2 \( 1 - 0.718iT - 2T^{2} \)
5 \( 1 + (-0.723 + 1.25i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.55 + 0.900i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.88 - 1.09i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.95 - 3.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.47 - 2.00i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.91 - 2.83i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (8.49 + 4.90i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.83iT - 31T^{2} \)
37 \( 1 + (0.411 + 0.713i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.90 + 10.2i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.76 - 6.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.33T + 47T^{2} \)
53 \( 1 + (0.996 + 0.575i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.79T + 59T^{2} \)
61 \( 1 + 2.35iT - 61T^{2} \)
67 \( 1 + 0.312T + 67T^{2} \)
71 \( 1 - 1.94iT - 71T^{2} \)
73 \( 1 + (-2.42 - 1.40i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 + (3.60 - 6.25i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.28 - 9.16i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.4 + 7.75i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.74081780839004887331188062812, −11.48113617974392786280267830376, −10.77744982421038846554442194964, −9.584028737515363849945203210082, −8.410857897967223870476793295058, −7.34871542426026103491656436754, −6.46837954440137944020105459179, −5.29024015107268419693774851830, −3.83610336315497493205073367140, −1.80078196486669761335350700544, 2.11320901146003072289620228849, 3.10202246782486130679339961163, 5.00663874624407385819696195196, 6.43567732523657629202681897234, 7.11683059679125641569615450876, 8.702285480052349881803607593195, 9.707962176898382737337131933139, 10.73515586825166345614143650832, 11.50302718051531675464774248105, 12.37464351701363521683564776030

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