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Properties

Label 2-189-63.16-c3-0-9
Degree $2$
Conductor $189$
Sign $0.408 + 0.912i$
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  + (−1.46 − 2.54i)2-s + (−0.305 + 0.529i)4-s + 3.69·5-s + (12.3 + 13.8i)7-s − 21.6·8-s + (−5.42 − 9.39i)10-s + 65.7·11-s + (2.73 + 4.74i)13-s + (16.9 − 51.6i)14-s + (34.2 + 59.3i)16-s + (25.1 + 43.5i)17-s + (0.769 − 1.33i)19-s + (−1.13 + 1.95i)20-s + (−96.4 − 166. i)22-s + 120.·23-s + ⋯
L(s)  = 1  + (−0.518 − 0.898i)2-s + (−0.0382 + 0.0662i)4-s + 0.330·5-s + (0.666 + 0.745i)7-s − 0.958·8-s + (−0.171 − 0.297i)10-s + 1.80·11-s + (0.0584 + 0.101i)13-s + (0.323 − 0.985i)14-s + (0.535 + 0.927i)16-s + (0.358 + 0.621i)17-s + (0.00929 − 0.0161i)19-s + (−0.0126 + 0.0218i)20-s + (−0.934 − 1.61i)22-s + 1.08·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.408 + 0.912i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.408 + 0.912i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.38583 - 0.898529i\)
\(L(\frac12)\) \(\approx\) \(1.38583 - 0.898529i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-12.3 - 13.8i)T \)
good2 \( 1 + (1.46 + 2.54i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 - 3.69T + 125T^{2} \)
11 \( 1 - 65.7T + 1.33e3T^{2} \)
13 \( 1 + (-2.73 - 4.74i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-25.1 - 43.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-0.769 + 1.33i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 120.T + 1.21e4T^{2} \)
29 \( 1 + (-39.2 + 68.0i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-151. + 262. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-96.6 + 167. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (196. + 340. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (138. - 239. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (126. + 218. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-204. - 353. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-131. + 227. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-56.1 - 97.2i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-49.1 + 85.1i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 255.T + 3.57e5T^{2} \)
73 \( 1 + (-344. - 596. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-542. - 939. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (152. - 263. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (550. - 953. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-493. + 855. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.69751251495017572083143558141, −11.13731978545647153627542158610, −9.845929040359828614752320850870, −9.197459969149440298021970428183, −8.277864250230503531988954619840, −6.56708239980144432539063224540, −5.62904755102712962432753809001, −3.91984740030551663445252150989, −2.29026629851707545551533713409, −1.18738431506747147830727440941, 1.20649344740253483569101141158, 3.39135055975917464676741133880, 4.91891328061850856596729761794, 6.42397391197071037018278869861, 7.05516374686578239427431601757, 8.189156683330253305508629579566, 9.072420062329041078934602187465, 10.03740587897522726151633793904, 11.43486089471500166614497992192, 12.06797070428074338310136905582

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