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Properties

Label 2-189-63.47-c3-0-20
Degree $2$
Conductor $189$
Sign $-0.141 + 0.990i$
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  + (4.53 − 2.61i)2-s + (9.72 − 16.8i)4-s + 12.2·5-s + (−10.8 − 15.0i)7-s − 59.9i·8-s + (55.3 − 31.9i)10-s + 63.6i·11-s + (13.7 − 7.96i)13-s + (−88.5 − 39.6i)14-s + (−79.2 − 137. i)16-s + (−17.9 − 31.0i)17-s + (−13.8 − 7.97i)19-s + (118. − 205. i)20-s + (166. + 288. i)22-s + 45.6i·23-s + ⋯
L(s)  = 1  + (1.60 − 0.926i)2-s + (1.21 − 2.10i)4-s + 1.09·5-s + (−0.585 − 0.810i)7-s − 2.64i·8-s + (1.75 − 1.01i)10-s + 1.74i·11-s + (0.294 − 0.169i)13-s + (−1.69 − 0.757i)14-s + (−1.23 − 2.14i)16-s + (−0.255 − 0.443i)17-s + (−0.166 − 0.0962i)19-s + (1.32 − 2.29i)20-s + (1.61 + 2.79i)22-s + 0.413i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 + 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.141 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.141 + 0.990i$
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: \((2,\ 189,\ (\ :3/2),\ -0.141 + 0.990i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.04641 - 3.51124i\)
\(L(\frac12)\) \(\approx\) \(3.04641 - 3.51124i\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.8 + 15.0i)T \)
good2 \( 1 + (-4.53 + 2.61i)T + (4 - 6.92i)T^{2} \)
5 \( 1 - 12.2T + 125T^{2} \)
11 \( 1 - 63.6iT - 1.33e3T^{2} \)
13 \( 1 + (-13.7 + 7.96i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (17.9 + 31.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (13.8 + 7.97i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 45.6iT - 1.21e4T^{2} \)
29 \( 1 + (-206. - 119. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (117. + 67.6i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-18.0 + 31.3i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-84.5 - 146. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (37.8 - 65.6i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (3.46 + 6.00i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-33.2 + 19.2i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-62.2 + 107. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (445. - 257. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (167. - 290. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 716. iT - 3.57e5T^{2} \)
73 \( 1 + (-725. + 418. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-204. - 354. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (135. - 234. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-717. + 1.24e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-433. - 250. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.19382690935871326032608269236, −10.95139501231869172676537144804, −10.09389147982300566974330915946, −9.541125874550993178399060485349, −7.14050687556851991998384703705, −6.26762823093407045508263164628, −5.08777892034395265671343505459, −4.12571958991029565118580894895, −2.71408783208862743329943644254, −1.52049661948517073201032883492, 2.50583896601720471665585538827, 3.62821495234324739384393300919, 5.20214192261290189003352902173, 6.12352457345911027466087953519, 6.40975696320538570895765196157, 8.151350355211648344933141813054, 9.080059087222062565404014438116, 10.67490966095685840500821378266, 11.85170544727137007621302864163, 12.74418258801691086148469340410

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