Properties

 Degree $2$ Conductor $38$ Sign $0.710 + 0.703i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−1.87 − 0.684i)2-s + (0.0408 + 0.231i)3-s + (3.06 + 2.57i)4-s + (8.45 − 7.09i)5-s + (0.0816 − 0.462i)6-s + (9.26 − 16.0i)7-s + (−4.00 − 6.92i)8-s + (25.3 − 9.21i)9-s + (−20.7 + 7.54i)10-s + (−6.98 − 12.1i)11-s + (−0.470 + 0.814i)12-s + (−9.37 + 53.1i)13-s + (−28.3 + 23.8i)14-s + (1.98 + 1.66i)15-s + (2.77 + 15.7i)16-s + (−7.65 − 2.78i)17-s + ⋯
 L(s)  = 1 + (−0.664 − 0.241i)2-s + (0.00785 + 0.0445i)3-s + (0.383 + 0.321i)4-s + (0.756 − 0.634i)5-s + (0.00555 − 0.0314i)6-s + (0.500 − 0.866i)7-s + (−0.176 − 0.306i)8-s + (0.937 − 0.341i)9-s + (−0.655 + 0.238i)10-s + (−0.191 − 0.331i)11-s + (−0.0113 + 0.0195i)12-s + (−0.199 + 1.13i)13-s + (−0.542 + 0.454i)14-s + (0.0342 + 0.0287i)15-s + (0.0434 + 0.246i)16-s + (−0.109 − 0.0397i)17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$38$$    =    $$2 \cdot 19$$ Sign: $0.710 + 0.703i$ Motivic weight: $$3$$ Character: $\chi_{38} (23, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 38,\ (\ :3/2),\ 0.710 + 0.703i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$1.02580 - 0.421862i$$ $$L(\frac12)$$ $$\approx$$ $$1.02580 - 0.421862i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.87 + 0.684i)T$$
19 $$1 + (31.6 - 76.5i)T$$
good3 $$1 + (-0.0408 - 0.231i)T + (-25.3 + 9.23i)T^{2}$$
5 $$1 + (-8.45 + 7.09i)T + (21.7 - 123. i)T^{2}$$
7 $$1 + (-9.26 + 16.0i)T + (-171.5 - 297. i)T^{2}$$
11 $$1 + (6.98 + 12.1i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 + (9.37 - 53.1i)T + (-2.06e3 - 751. i)T^{2}$$
17 $$1 + (7.65 + 2.78i)T + (3.76e3 + 3.15e3i)T^{2}$$
23 $$1 + (93.7 + 78.6i)T + (2.11e3 + 1.19e4i)T^{2}$$
29 $$1 + (-208. + 75.7i)T + (1.86e4 - 1.56e4i)T^{2}$$
31 $$1 + (23.1 - 40.0i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + 99.6T + 5.06e4T^{2}$$
41 $$1 + (-55.1 - 312. i)T + (-6.47e4 + 2.35e4i)T^{2}$$
43 $$1 + (358. - 300. i)T + (1.38e4 - 7.82e4i)T^{2}$$
47 $$1 + (-16.4 + 5.96i)T + (7.95e4 - 6.67e4i)T^{2}$$
53 $$1 + (-33.9 - 28.4i)T + (2.58e4 + 1.46e5i)T^{2}$$
59 $$1 + (599. + 218. i)T + (1.57e5 + 1.32e5i)T^{2}$$
61 $$1 + (-631. - 529. i)T + (3.94e4 + 2.23e5i)T^{2}$$
67 $$1 + (-789. + 287. i)T + (2.30e5 - 1.93e5i)T^{2}$$
71 $$1 + (442. - 371. i)T + (6.21e4 - 3.52e5i)T^{2}$$
73 $$1 + (131. + 744. i)T + (-3.65e5 + 1.33e5i)T^{2}$$
79 $$1 + (-58.9 - 334. i)T + (-4.63e5 + 1.68e5i)T^{2}$$
83 $$1 + (-382. + 663. i)T + (-2.85e5 - 4.95e5i)T^{2}$$
89 $$1 + (-157. + 893. i)T + (-6.62e5 - 2.41e5i)T^{2}$$
97 $$1 + (1.58e3 + 575. i)T + (6.99e5 + 5.86e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$