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Properties

Label 2-189-63.47-c3-0-9
Degree $2$
Conductor $189$
Sign $0.845 + 0.533i$
Analytic cond. $11.1513$
Root an. cond. $3.33936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.39 + 2.53i)2-s + (8.88 − 15.3i)4-s − 8.42·5-s + (−13.4 + 12.7i)7-s + 49.6i·8-s + (37.0 − 21.3i)10-s + 45.8i·11-s + (−18.3 + 10.6i)13-s + (26.8 − 90.1i)14-s + (−54.8 − 95.0i)16-s + (−8.26 − 14.3i)17-s + (−49.7 − 28.7i)19-s + (−74.8 + 129. i)20-s + (−116. − 201. i)22-s − 193. i·23-s + ⋯
L(s)  = 1  + (−1.55 + 0.897i)2-s + (1.11 − 1.92i)4-s − 0.753·5-s + (−0.726 + 0.687i)7-s + 2.19i·8-s + (1.17 − 0.676i)10-s + 1.25i·11-s + (−0.392 + 0.226i)13-s + (0.512 − 1.72i)14-s + (−0.857 − 1.48i)16-s + (−0.117 − 0.204i)17-s + (−0.600 − 0.346i)19-s + (−0.837 + 1.45i)20-s + (−1.12 − 1.95i)22-s − 1.75i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.845 + 0.533i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.845 + 0.533i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.254914 - 0.0737134i\)
\(L(\frac12)\) \(\approx\) \(0.254914 - 0.0737134i\)
\(L(\frac{5}{2})\) 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 \( 1 + (13.4 - 12.7i)T \)
good2 \( 1 + (4.39 - 2.53i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + 8.42T + 125T^{2} \)
11 \( 1 - 45.8iT - 1.33e3T^{2} \)
13 \( 1 + (18.3 - 10.6i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (8.26 + 14.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (49.7 + 28.7i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 193. iT - 1.21e4T^{2} \)
29 \( 1 + (-47.1 - 27.2i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-254. - 147. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-185. + 320. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-166. - 287. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-104. + 180. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (209. + 363. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (275. - 159. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-149. + 258. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (195. - 113. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-49.8 + 86.3i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 176. iT - 3.57e5T^{2} \)
73 \( 1 + (-142. + 82.2i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (187. + 324. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-457. + 792. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-67.8 + 117. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-377. - 217. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.89065308063989348080265670339, −10.60413250918183150041030852679, −9.758418034005919850538647527985, −8.923338569055138957161220600484, −8.022328270154670694063771074624, −7.00450685523983211912496510106, −6.27992550799072417078950220550, −4.63505011476697814706002765582, −2.34071512940536099122655038040, −0.25351125039679061165197714223, 0.950419673788499392376246727693, 2.90301704757884380023609184071, 3.91441163442357623457885958766, 6.28697585878974210980232030281, 7.66372993430272512891251360517, 8.169226021162898891162907607668, 9.407883345013517528169290102229, 10.14217205746585222686447414201, 11.14881760491498230569692219694, 11.70552315683949324330396447854

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