# Properties

 Degree $2$ Conductor $38$ Sign $-0.999 - 0.0387i$ Motivic weight $4$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.82i·2-s − 7.80i·3-s − 8.00·4-s − 33.0·5-s − 22.0·6-s + 16.0·7-s + 22.6i·8-s + 20.1·9-s + 93.6i·10-s − 215.·11-s + 62.4i·12-s − 281. i·13-s − 45.4i·14-s + 258. i·15-s + 64.0·16-s + 226.·17-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.867i·3-s − 0.500·4-s − 1.32·5-s − 0.613·6-s + 0.328·7-s + 0.353i·8-s + 0.248·9-s + 0.936i·10-s − 1.78·11-s + 0.433i·12-s − 1.66i·13-s − 0.232i·14-s + 1.14i·15-s + 0.250·16-s + 0.784·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0387i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$38$$    =    $$2 \cdot 19$$ Sign: $-0.999 - 0.0387i$ Motivic weight: $$4$$ Character: $\chi_{38} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 38,\ (\ :2),\ -0.999 - 0.0387i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$0.0155537 + 0.802003i$$ $$L(\frac12)$$ $$\approx$$ $$0.0155537 + 0.802003i$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 2.82iT$$
19 $$1 + (-13.9 + 360. i)T$$
good3 $$1 + 7.80iT - 81T^{2}$$
5 $$1 + 33.0T + 625T^{2}$$
7 $$1 - 16.0T + 2.40e3T^{2}$$
11 $$1 + 215.T + 1.46e4T^{2}$$
13 $$1 + 281. iT - 2.85e4T^{2}$$
17 $$1 - 226.T + 8.35e4T^{2}$$
23 $$1 - 414.T + 2.79e5T^{2}$$
29 $$1 - 606. iT - 7.07e5T^{2}$$
31 $$1 + 478. iT - 9.23e5T^{2}$$
37 $$1 + 104. iT - 1.87e6T^{2}$$
41 $$1 - 1.89e3iT - 2.82e6T^{2}$$
43 $$1 - 315.T + 3.41e6T^{2}$$
47 $$1 + 474.T + 4.87e6T^{2}$$
53 $$1 + 774. iT - 7.89e6T^{2}$$
59 $$1 - 567. iT - 1.21e7T^{2}$$
61 $$1 - 4.79e3T + 1.38e7T^{2}$$
67 $$1 - 4.46e3iT - 2.01e7T^{2}$$
71 $$1 + 7.63e3iT - 2.54e7T^{2}$$
73 $$1 - 8.39e3T + 2.83e7T^{2}$$
79 $$1 + 9.70e3iT - 3.89e7T^{2}$$
83 $$1 + 1.05e4T + 4.74e7T^{2}$$
89 $$1 + 1.02e4iT - 6.27e7T^{2}$$
97 $$1 - 1.54e4iT - 8.85e7T^{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

−15.02903363063193344973352300937, −13.14636067384833494790865700666, −12.69364924558045831202186445849, −11.38555878630890379595118880875, −10.31979767975201805991687228674, −8.129153141515518737636400381076, −7.54111090346297570572808495039, −5.04864726349963879376606642930, −2.99676737039772214886529090026, −0.56051810683536469085704849293, 3.89393897250238198753432320968, 5.07252253047712024793266339879, 7.28164823474562927200422926353, 8.317171134728054257926125755522, 9.887664685539706176412332400175, 11.17891234321650548528556370268, 12.56538116149668367564998660141, 14.19320065165023623846737036370, 15.34603367814436935349406273368, 15.95466063782112133522198189894