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Properties

Label 2-189-63.59-c3-0-14
Degree $2$
Conductor $189$
Sign $0.241 - 0.970i$
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  + (4.26 + 2.46i)2-s + (8.14 + 14.1i)4-s + 9.24·5-s + (17.8 − 4.75i)7-s + 40.8i·8-s + (39.4 + 22.7i)10-s + 15.2i·11-s + (−61.0 − 35.2i)13-s + (88.1 + 23.7i)14-s + (−35.5 + 61.5i)16-s + (−17.8 + 30.9i)17-s + (91.6 − 52.9i)19-s + (75.3 + 130. i)20-s + (−37.6 + 65.1i)22-s + 92.4i·23-s + ⋯
L(s)  = 1  + (1.50 + 0.871i)2-s + (1.01 + 1.76i)4-s + 0.827·5-s + (0.966 − 0.256i)7-s + 1.80i·8-s + (1.24 + 0.720i)10-s + 0.418i·11-s + (−1.30 − 0.752i)13-s + (1.68 + 0.454i)14-s + (−0.555 + 0.961i)16-s + (−0.255 + 0.441i)17-s + (1.10 − 0.638i)19-s + (0.842 + 1.45i)20-s + (−0.364 + 0.631i)22-s + 0.838i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.241 - 0.970i$
Analytic conductor: \(11.1513\)
Root analytic conductor: \(3.33936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :3/2),\ 0.241 - 0.970i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.62968 + 2.83760i\)
\(L(\frac12)\) \(\approx\) \(3.62968 + 2.83760i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-17.8 + 4.75i)T \)
good2 \( 1 + (-4.26 - 2.46i)T + (4 + 6.92i)T^{2} \)
5 \( 1 - 9.24T + 125T^{2} \)
11 \( 1 - 15.2iT - 1.33e3T^{2} \)
13 \( 1 + (61.0 + 35.2i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (17.8 - 30.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-91.6 + 52.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 92.4iT - 1.21e4T^{2} \)
29 \( 1 + (222. - 128. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (123. - 71.3i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (100. + 173. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-162. + 280. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (92.5 + 160. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-235. + 408. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-247. - 142. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (85.0 + 147. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-222. - 128. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (501. + 868. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 445. iT - 3.57e5T^{2} \)
73 \( 1 + (81.3 + 46.9i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (231. - 401. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-348. - 603. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-338. - 586. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-402. + 232. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.62453659010089183069143748985, −11.75430643958663740337971984198, −10.52925767488638430089189096284, −9.226094845168084657552324784536, −7.59303734071495881674010967914, −7.14114162438379769735967689116, −5.44691755252945255049746461055, −5.23664838923930781046914999588, −3.76356183787039868476273399698, −2.15909938438262689644628149106, 1.68726341771517330657270493783, 2.66359748628411796405119787208, 4.28664938810452599537231338417, 5.23310989938092787533788881783, 6.06190105854422216510357685853, 7.57935183397348251321922555490, 9.296357902513085501198647153927, 10.20187271032121889119880161966, 11.42348147455222867439858010184, 11.82058857959828158757786540387

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