# Properties

 Degree 2 Conductor $3^{3}$ Sign $0.0270 - 0.999i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−4.97 − 1.81i)2-s + (−3.67 − 3.67i)3-s + (15.3 + 12.9i)4-s + (−1.82 + 10.3i)5-s + (11.6 + 24.9i)6-s + (−5.92 + 4.96i)7-s + (−31.9 − 55.3i)8-s + (0.0364 + 26.9i)9-s + (27.8 − 48.2i)10-s + (2.99 + 16.9i)11-s + (−9.15 − 103. i)12-s + (−67.4 + 24.5i)13-s + (38.4 − 14.0i)14-s + (44.7 − 31.3i)15-s + (30.9 + 175. i)16-s + (−23.3 + 40.4i)17-s + ⋯
 L(s)  = 1 + (−1.76 − 0.640i)2-s + (−0.707 − 0.706i)3-s + (1.92 + 1.61i)4-s + (−0.163 + 0.925i)5-s + (0.792 + 1.69i)6-s + (−0.319 + 0.268i)7-s + (−1.41 − 2.44i)8-s + (0.00134 + 0.999i)9-s + (0.880 − 1.52i)10-s + (0.0820 + 0.465i)11-s + (−0.220 − 2.49i)12-s + (−1.43 + 0.523i)13-s + (0.734 − 0.267i)14-s + (0.769 − 0.539i)15-s + (0.483 + 2.74i)16-s + (−0.333 + 0.577i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0270 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0270 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$27$$    =    $$3^{3}$$ $$\varepsilon$$ = $0.0270 - 0.999i$ motivic weight = $$3$$ character : $\chi_{27} (4, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 27,\ (\ :3/2),\ 0.0270 - 0.999i)$ $L(2)$ $\approx$ $0.142034 + 0.138241i$ $L(\frac12)$ $\approx$ $0.142034 + 0.138241i$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 3$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 + (3.67 + 3.67i)T$$
good2 $$1 + (4.97 + 1.81i)T + (6.12 + 5.14i)T^{2}$$
5 $$1 + (1.82 - 10.3i)T + (-117. - 42.7i)T^{2}$$
7 $$1 + (5.92 - 4.96i)T + (59.5 - 337. i)T^{2}$$
11 $$1 + (-2.99 - 16.9i)T + (-1.25e3 + 455. i)T^{2}$$
13 $$1 + (67.4 - 24.5i)T + (1.68e3 - 1.41e3i)T^{2}$$
17 $$1 + (23.3 - 40.4i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (21.3 + 36.9i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (-36.3 - 30.5i)T + (2.11e3 + 1.19e4i)T^{2}$$
29 $$1 + (139. + 50.8i)T + (1.86e4 + 1.56e4i)T^{2}$$
31 $$1 + (120. + 101. i)T + (5.17e3 + 2.93e4i)T^{2}$$
37 $$1 + (-35.3 + 61.2i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 + (219. - 79.8i)T + (5.27e4 - 4.43e4i)T^{2}$$
43 $$1 + (-71.5 - 405. i)T + (-7.47e4 + 2.71e4i)T^{2}$$
47 $$1 + (-327. + 275. i)T + (1.80e4 - 1.02e5i)T^{2}$$
53 $$1 - 43.9T + 1.48e5T^{2}$$
59 $$1 + (57.9 - 328. i)T + (-1.92e5 - 7.02e4i)T^{2}$$
61 $$1 + (-292. + 245. i)T + (3.94e4 - 2.23e5i)T^{2}$$
67 $$1 + (230. - 83.9i)T + (2.30e5 - 1.93e5i)T^{2}$$
71 $$1 + (-403. + 698. i)T + (-1.78e5 - 3.09e5i)T^{2}$$
73 $$1 + (-175. - 303. i)T + (-1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (-9.99e2 - 363. i)T + (3.77e5 + 3.16e5i)T^{2}$$
83 $$1 + (959. + 349. i)T + (4.38e5 + 3.67e5i)T^{2}$$
89 $$1 + (-320. - 555. i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 + (-60.5 - 343. i)T + (-8.57e5 + 3.12e5i)T^{2}$$
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\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}

## Imaginary part of the first few zeros on the critical line

−17.42554496482041569484900563876, −16.67368131288198248801047995580, −15.08042784199882822683271098189, −12.71722549467447083248042933156, −11.62975354132174677959807218031, −10.69664984093828308744753164082, −9.434471499082734571318383716230, −7.60684881628411948000616370884, −6.73368558866435084205526898646, −2.30508093884605609361262026822, 0.34709593077116122194215721108, 5.34841752229001997160381366933, 7.06965634178584925744342632286, 8.720406020343865834160935969782, 9.736481879280681309870817747118, 10.82066414227898608831399278589, 12.26822145856954319916109403850, 14.85529326700521138282685391270, 15.98113181031536309994430315961, 16.75643505204580486783296742661