# Properties

 Degree 2 Conductor 43 Sign $0.883 - 0.468i$ Motivic weight 6 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.178i·2-s + (33.4 + 19.3i)3-s + 63.9·4-s + (21.1 + 12.1i)5-s + (−3.45 + 5.97i)6-s + (409. − 236. i)7-s + 22.8i·8-s + (380. + 659. i)9-s + (−2.17 + 3.77i)10-s − 2.21e3·11-s + (2.13e3 + 1.23e3i)12-s + (−1.29e3 − 2.24e3i)13-s + (42.2 + 73.1i)14-s + (470. + 814. i)15-s + 4.08e3·16-s + (4.48e3 + 7.76e3i)17-s + ⋯
 L(s)  = 1 + 0.0223i·2-s + (1.23 + 0.714i)3-s + 0.999·4-s + (0.168 + 0.0974i)5-s + (−0.0159 + 0.0276i)6-s + (1.19 − 0.688i)7-s + 0.0447i·8-s + (0.522 + 0.904i)9-s + (−0.00217 + 0.00377i)10-s − 1.66·11-s + (1.23 + 0.714i)12-s + (−0.590 − 1.02i)13-s + (0.0153 + 0.0266i)14-s + (0.139 + 0.241i)15-s + 0.998·16-s + (0.912 + 1.58i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $0.883 - 0.468i$ motivic weight = $$6$$ character : $\chi_{43} (37, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :3),\ 0.883 - 0.468i)$$ $$L(\frac{7}{2})$$ $$\approx$$ $$3.12515 + 0.777646i$$ $$L(\frac12)$$ $$\approx$$ $$3.12515 + 0.777646i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 43$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad43 $$1 + (4.31e4 + 6.67e4i)T$$
good2 $$1 - 0.178iT - 64T^{2}$$
3 $$1 + (-33.4 - 19.3i)T + (364.5 + 631. i)T^{2}$$
5 $$1 + (-21.1 - 12.1i)T + (7.81e3 + 1.35e4i)T^{2}$$
7 $$1 + (-409. + 236. i)T + (5.88e4 - 1.01e5i)T^{2}$$
11 $$1 + 2.21e3T + 1.77e6T^{2}$$
13 $$1 + (1.29e3 + 2.24e3i)T + (-2.41e6 + 4.18e6i)T^{2}$$
17 $$1 + (-4.48e3 - 7.76e3i)T + (-1.20e7 + 2.09e7i)T^{2}$$
19 $$1 + (699. + 403. i)T + (2.35e7 + 4.07e7i)T^{2}$$
23 $$1 + (2.74e3 - 4.74e3i)T + (-7.40e7 - 1.28e8i)T^{2}$$
29 $$1 + (1.47e4 - 8.52e3i)T + (2.97e8 - 5.15e8i)T^{2}$$
31 $$1 + (2.25e4 - 3.90e4i)T + (-4.43e8 - 7.68e8i)T^{2}$$
37 $$1 + (8.88e3 + 5.12e3i)T + (1.28e9 + 2.22e9i)T^{2}$$
41 $$1 + 1.69e4T + 4.75e9T^{2}$$
47 $$1 - 7.30e4T + 1.07e10T^{2}$$
53 $$1 + (-9.62e4 + 1.66e5i)T + (-1.10e10 - 1.91e10i)T^{2}$$
59 $$1 + 1.37e5T + 4.21e10T^{2}$$
61 $$1 + (3.54e5 - 2.04e5i)T + (2.57e10 - 4.46e10i)T^{2}$$
67 $$1 + (-1.59e5 + 2.75e5i)T + (-4.52e10 - 7.83e10i)T^{2}$$
71 $$1 + (-4.49e5 + 2.59e5i)T + (6.40e10 - 1.10e11i)T^{2}$$
73 $$1 + (-1.89e5 + 1.09e5i)T + (7.56e10 - 1.31e11i)T^{2}$$
79 $$1 + (-3.84e5 - 6.66e5i)T + (-1.21e11 + 2.10e11i)T^{2}$$
83 $$1 + (-1.26e5 + 2.18e5i)T + (-1.63e11 - 2.83e11i)T^{2}$$
89 $$1 + (4.34e5 + 2.51e5i)T + (2.48e11 + 4.30e11i)T^{2}$$
97 $$1 - 2.80e5T + 8.32e11T^{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

−14.95824721873843101440194035004, −13.98487458790404620409537615809, −12.51901982247667929777265336324, −10.61067089388471424241547919877, −10.31092129749443475297924738501, −8.172701466475694140904071684820, −7.67247385357316396059843256589, −5.33257664097387648681427007021, −3.43541113094124009796886098892, −2.03265488298720026393638417384, 1.89713954036931418794631167520, 2.67876970017467860119020730749, 5.35418517349218127328495169393, 7.41694769761774211374716682064, 7.947848976466934167765325026974, 9.475060356794030300901803199353, 11.21268310598385895752557853997, 12.25794048684466737669665770913, 13.59081112095010774188232223720, 14.60430045395242671850211306215