# Properties

 Degree $2$ Conductor $38$ Sign $0.136 + 0.990i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (3.06 + 2.57i)2-s + (−24.7 + 9.01i)3-s + (2.77 + 15.7i)4-s + (14.6 − 83.2i)5-s + (−99.1 − 36.0i)6-s + (−7.63 − 13.2i)7-s + (−32.0 + 55.4i)8-s + (346. − 290. i)9-s + (258. − 217. i)10-s + (225. − 390. i)11-s + (−210. − 365. i)12-s + (−754. − 274. i)13-s + (10.5 − 60.1i)14-s + (386. + 2.19e3i)15-s + (−240. + 87.5i)16-s + (−1.01e3 − 855. i)17-s + ⋯
 L(s)  = 1 + (0.541 + 0.454i)2-s + (−1.58 + 0.578i)3-s + (0.0868 + 0.492i)4-s + (0.262 − 1.48i)5-s + (−1.12 − 0.409i)6-s + (−0.0588 − 0.101i)7-s + (−0.176 + 0.306i)8-s + (1.42 − 1.19i)9-s + (0.819 − 0.687i)10-s + (0.561 − 0.971i)11-s + (−0.422 − 0.732i)12-s + (−1.23 − 0.450i)13-s + (0.0144 − 0.0819i)14-s + (0.444 + 2.51i)15-s + (−0.234 + 0.0855i)16-s + (−0.855 − 0.717i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$38$$    =    $$2 \cdot 19$$ Sign: $0.136 + 0.990i$ Motivic weight: $$5$$ Character: $\chi_{38} (35, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 38,\ (\ :5/2),\ 0.136 + 0.990i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.588287 - 0.512851i$$ $$L(\frac12)$$ $$\approx$$ $$0.588287 - 0.512851i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-3.06 - 2.57i)T$$
19 $$1 + (1.50e3 - 459. i)T$$
good3 $$1 + (24.7 - 9.01i)T + (186. - 156. i)T^{2}$$
5 $$1 + (-14.6 + 83.2i)T + (-2.93e3 - 1.06e3i)T^{2}$$
7 $$1 + (7.63 + 13.2i)T + (-8.40e3 + 1.45e4i)T^{2}$$
11 $$1 + (-225. + 390. i)T + (-8.05e4 - 1.39e5i)T^{2}$$
13 $$1 + (754. + 274. i)T + (2.84e5 + 2.38e5i)T^{2}$$
17 $$1 + (1.01e3 + 855. i)T + (2.46e5 + 1.39e6i)T^{2}$$
23 $$1 + (-65.9 - 373. i)T + (-6.04e6 + 2.20e6i)T^{2}$$
29 $$1 + (-6.38e3 + 5.35e3i)T + (3.56e6 - 2.01e7i)T^{2}$$
31 $$1 + (-3.34e3 - 5.79e3i)T + (-1.43e7 + 2.47e7i)T^{2}$$
37 $$1 + 4.85e3T + 6.93e7T^{2}$$
41 $$1 + (1.18e4 - 4.30e3i)T + (8.87e7 - 7.44e7i)T^{2}$$
43 $$1 + (-2.55e3 + 1.45e4i)T + (-1.38e8 - 5.02e7i)T^{2}$$
47 $$1 + (9.45e3 - 7.92e3i)T + (3.98e7 - 2.25e8i)T^{2}$$
53 $$1 + (-286. - 1.62e3i)T + (-3.92e8 + 1.43e8i)T^{2}$$
59 $$1 + (-4.24e3 - 3.55e3i)T + (1.24e8 + 7.04e8i)T^{2}$$
61 $$1 + (-5.83e3 - 3.31e4i)T + (-7.93e8 + 2.88e8i)T^{2}$$
67 $$1 + (-824. + 691. i)T + (2.34e8 - 1.32e9i)T^{2}$$
71 $$1 + (-4.28e3 + 2.42e4i)T + (-1.69e9 - 6.17e8i)T^{2}$$
73 $$1 + (-4.67e4 + 1.70e4i)T + (1.58e9 - 1.33e9i)T^{2}$$
79 $$1 + (2.89e4 - 1.05e4i)T + (2.35e9 - 1.97e9i)T^{2}$$
83 $$1 + (1.88e4 + 3.25e4i)T + (-1.96e9 + 3.41e9i)T^{2}$$
89 $$1 + (-8.71e4 - 3.17e4i)T + (4.27e9 + 3.58e9i)T^{2}$$
97 $$1 + (8.78e4 + 7.37e4i)T + (1.49e9 + 8.45e9i)T^{2}$$
show less
$$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

−15.45722533613168473495995629542, −13.65976167669869281400378221855, −12.40804618436842582355030304218, −11.72784761793478615161570717778, −10.17150156461963839842113773090, −8.672560494218737899361745267430, −6.51648878261310160418284438603, −5.28475417705690817684029230914, −4.48476188896600499318283939100, −0.44145639808686111498609112641, 2.13646882255354498416696226385, 4.67826582885240886180305684776, 6.44240924159415283368875183907, 6.89976819340604751811454783918, 10.04286037450898769593208720836, 10.90944112396902150549610708413, 11.90535475653944478446556014140, 12.80340514078238823927251138414, 14.33448749049354330896140947097, 15.31759380163255995715695640221