# Properties

 Degree 2 Conductor 43 Sign $-0.776 - 0.630i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.75i·2-s + 12.3i·3-s + 8.40·4-s + 21.9i·5-s − 33.9·6-s − 63.3i·7-s + 67.2i·8-s − 71.0·9-s − 60.5·10-s − 1.17·11-s + 103. i·12-s − 173.·13-s + 174.·14-s − 271.·15-s − 51.0·16-s + 469.·17-s + ⋯
 L(s)  = 1 + 0.689i·2-s + 1.37i·3-s + 0.525·4-s + 0.879i·5-s − 0.944·6-s − 1.29i·7-s + 1.05i·8-s − 0.877·9-s − 0.605·10-s − 0.00967·11-s + 0.719i·12-s − 1.02·13-s + 0.890·14-s − 1.20·15-s − 0.199·16-s + 1.62·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $-0.776 - 0.630i$ motivic weight = $$4$$ character : $\chi_{43} (42, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :2),\ -0.776 - 0.630i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$0.557954 + 1.57290i$$ $$L(\frac12)$$ $$\approx$$ $$0.557954 + 1.57290i$$ $$L(3)$$ 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 + (1.43e3 + 1.16e3i)T$$
good2 $$1 - 2.75iT - 16T^{2}$$
3 $$1 - 12.3iT - 81T^{2}$$
5 $$1 - 21.9iT - 625T^{2}$$
7 $$1 + 63.3iT - 2.40e3T^{2}$$
11 $$1 + 1.17T + 1.46e4T^{2}$$
13 $$1 + 173.T + 2.85e4T^{2}$$
17 $$1 - 469.T + 8.35e4T^{2}$$
19 $$1 + 27.8iT - 1.30e5T^{2}$$
23 $$1 - 79.3T + 2.79e5T^{2}$$
29 $$1 + 696. iT - 7.07e5T^{2}$$
31 $$1 - 1.19e3T + 9.23e5T^{2}$$
37 $$1 - 1.53e3iT - 1.87e6T^{2}$$
41 $$1 + 2.45e3T + 2.82e6T^{2}$$
47 $$1 - 1.69e3T + 4.87e6T^{2}$$
53 $$1 + 1.99e3T + 7.89e6T^{2}$$
59 $$1 - 3.56e3T + 1.21e7T^{2}$$
61 $$1 + 5.44e3iT - 1.38e7T^{2}$$
67 $$1 - 4.93e3T + 2.01e7T^{2}$$
71 $$1 + 9.66e3iT - 2.54e7T^{2}$$
73 $$1 + 5.60e3iT - 2.83e7T^{2}$$
79 $$1 + 1.14e4T + 3.89e7T^{2}$$
83 $$1 - 5.52e3T + 4.74e7T^{2}$$
89 $$1 + 2.55e3iT - 6.27e7T^{2}$$
97 $$1 - 3.65e3T + 8.85e7T^{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

−15.55341630590062958172333336082, −14.78979826804969102890607750265, −13.98904015320510274380246628740, −11.74919971998207822593983085432, −10.47636010720746538350222720022, −9.994771621373234759823917643754, −7.84037387907704472559513336440, −6.73123257422785286314077517651, −4.96499534700002864458523298426, −3.26704972758665571293883281250, 1.22828245147751661481197054843, 2.62780511990073890195166163342, 5.51993191471330613894584971890, 7.04123679637500052230556178248, 8.367753879461688962762470226169, 9.880935316537476201243816783775, 11.82387953545417243493547449506, 12.27838502056889228197313882025, 12.94442810054998090173961859859, 14.60894644641096190324918832258