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Properties

Label 2-189-63.47-c3-0-13
Degree $2$
Conductor $189$
Sign $0.997 + 0.0649i$
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  + (−4.66 + 2.69i)2-s + (10.4 − 18.1i)4-s + 11.1·5-s + (18.4 − 0.902i)7-s + 69.7i·8-s + (−52.0 + 30.0i)10-s − 20.9i·11-s + (46.0 − 26.5i)13-s + (−83.7 + 53.9i)14-s + (−103. − 179. i)16-s + (−44.2 − 76.6i)17-s + (−40.4 − 23.3i)19-s + (116. − 202. i)20-s + (56.2 + 97.4i)22-s − 78.6i·23-s + ⋯
L(s)  = 1  + (−1.64 + 0.951i)2-s + (1.30 − 2.26i)4-s + 0.998·5-s + (0.998 − 0.0487i)7-s + 3.08i·8-s + (−1.64 + 0.949i)10-s − 0.573i·11-s + (0.981 − 0.566i)13-s + (−1.59 + 1.03i)14-s + (−1.62 − 2.80i)16-s + (−0.631 − 1.09i)17-s + (−0.488 − 0.281i)19-s + (1.30 − 2.26i)20-s + (0.545 + 0.944i)22-s − 0.712i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0649i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.997 + 0.0649i$
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: \((2,\ 189,\ (\ :3/2),\ 0.997 + 0.0649i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.00460 - 0.0326457i\)
\(L(\frac12)\) \(\approx\) \(1.00460 - 0.0326457i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.4 + 0.902i)T \)
good2 \( 1 + (4.66 - 2.69i)T + (4 - 6.92i)T^{2} \)
5 \( 1 - 11.1T + 125T^{2} \)
11 \( 1 + 20.9iT - 1.33e3T^{2} \)
13 \( 1 + (-46.0 + 26.5i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (44.2 + 76.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (40.4 + 23.3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 78.6iT - 1.21e4T^{2} \)
29 \( 1 + (-17.2 - 9.97i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-24.2 - 14.0i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (122. - 211. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-16.9 - 29.3i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-87.5 + 151. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (73.4 + 127. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-656. + 378. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-41.9 + 72.6i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (79.8 - 46.0i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-448. + 776. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 706. iT - 3.57e5T^{2} \)
73 \( 1 + (529. - 305. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-133. - 230. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (311. - 539. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-355. + 615. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-498. - 287. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.50567971527712024417650916089, −10.74586045804402156876446462138, −9.943793677828324067250484998956, −8.783535489481924566058471804855, −8.350124175765806484327006842624, −7.05675227843837703666227899199, −6.09371700395308586283931078493, −5.14257039501470084599170193347, −2.17556966452660133367450401833, −0.825224664883407266031444315344, 1.46912226211242076686438803310, 2.15599121265724831613753002029, 4.03693842384485655166150167257, 6.08676892103480481937173612235, 7.40047390821768571718460423999, 8.497856969918005734574361336567, 9.131722235781081377570539493646, 10.19906459285211908068697518386, 10.87183216790594948993788781450, 11.70899984766251203092695970355

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