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Properties

Label 2-189-63.47-c3-0-17
Degree $2$
Conductor $189$
Sign $0.00591 + 0.999i$
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  + (3.22 − 1.86i)2-s + (2.92 − 5.06i)4-s − 2.66·5-s + (9.53 − 15.8i)7-s + 8.01i·8-s + (−8.57 + 4.94i)10-s − 67.9i·11-s + (69.6 − 40.2i)13-s + (1.16 − 68.9i)14-s + (38.2 + 66.3i)16-s + (−29.6 − 51.2i)17-s + (−5.86 − 3.38i)19-s + (−7.77 + 13.4i)20-s + (−126. − 219. i)22-s + 126. i·23-s + ⋯
L(s)  = 1  + (1.13 − 0.657i)2-s + (0.365 − 0.632i)4-s − 0.237·5-s + (0.514 − 0.857i)7-s + 0.354i·8-s + (−0.271 + 0.156i)10-s − 1.86i·11-s + (1.48 − 0.858i)13-s + (0.0223 − 1.31i)14-s + (0.598 + 1.03i)16-s + (−0.422 − 0.731i)17-s + (−0.0707 − 0.0408i)19-s + (−0.0869 + 0.150i)20-s + (−1.22 − 2.12i)22-s + 1.14i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00591 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.00591 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.00591 + 0.999i$
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: \((2,\ 189,\ (\ :3/2),\ 0.00591 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.20258 - 2.18960i\)
\(L(\frac12)\) \(\approx\) \(2.20258 - 2.18960i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.53 + 15.8i)T \)
good2 \( 1 + (-3.22 + 1.86i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + 2.66T + 125T^{2} \)
11 \( 1 + 67.9iT - 1.33e3T^{2} \)
13 \( 1 + (-69.6 + 40.2i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (29.6 + 51.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (5.86 + 3.38i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 126. iT - 1.21e4T^{2} \)
29 \( 1 + (-114. - 66.0i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-62.8 - 36.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (63.8 - 110. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (4.93 + 8.54i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (108. - 188. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-92.9 - 160. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-174. + 100. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-89.2 + 154. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (195. - 112. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (219. - 379. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 533. iT - 3.57e5T^{2} \)
73 \( 1 + (-287. + 165. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-503. - 872. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-590. + 1.02e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (66.3 - 114. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (761. + 439. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.64968094902895803481255965542, −11.23358506893931177698822309254, −10.45177290840761527456744643492, −8.652495681886413854778908487952, −7.905655480696401437840547755015, −6.21122611313465897546963389192, −5.21317196814948427815589346355, −3.86888672754010364241880109342, −3.14483622804952750832301166782, −1.07648986133207894248342625043, 1.99880840856914245737795220200, 3.99608705559730266486354604114, 4.72028713476111324855035259865, 6.01684620731449802224847848908, 6.82170511379739838816346868569, 8.105907161294207571619166583432, 9.224895436715253166688880991677, 10.48853749336337099500523042500, 11.85577378953554951290183207386, 12.41904430694587807045791887507

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