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Properties

Label 2-189-63.59-c3-0-7
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: $\chi_{189} (17, \cdot )$
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

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

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