Properties

 Degree 2 Conductor $2^{6} \cdot 5$ Sign $0.347 + 0.937i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 − 11.4·3-s + (−32.3 − 45.6i)5-s − 121. i·7-s − 113.·9-s + 625. i·11-s + 924.·13-s + (368. + 520. i)15-s + 1.32e3i·17-s − 1.13e3i·19-s + 1.38e3i·21-s + 3.05e3i·23-s + (−1.03e3 + 2.94e3i)25-s + 4.05e3·27-s − 8.23e3i·29-s + 1.98e3·31-s + ⋯
 L(s)  = 1 − 0.731·3-s + (−0.578 − 0.816i)5-s − 0.936i·7-s − 0.465·9-s + 1.55i·11-s + 1.51·13-s + (0.422 + 0.596i)15-s + 1.11i·17-s − 0.721i·19-s + 0.684i·21-s + 1.20i·23-s + (−0.331 + 0.943i)25-s + 1.07·27-s − 1.81i·29-s + 0.371·31-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$320$$    =    $$2^{6} \cdot 5$$ $$\varepsilon$$ = $0.347 + 0.937i$ motivic weight = $$5$$ character : $\chi_{320} (289, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 320,\ (\ :5/2),\ 0.347 + 0.937i)$$ $$L(3)$$ $$\approx$$ $$1.078375237$$ $$L(\frac12)$$ $$\approx$$ $$1.078375237$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;5\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + (32.3 + 45.6i)T$$
good3 $$1 + 11.4T + 243T^{2}$$
7 $$1 + 121. iT - 1.68e4T^{2}$$
11 $$1 - 625. iT - 1.61e5T^{2}$$
13 $$1 - 924.T + 3.71e5T^{2}$$
17 $$1 - 1.32e3iT - 1.41e6T^{2}$$
19 $$1 + 1.13e3iT - 2.47e6T^{2}$$
23 $$1 - 3.05e3iT - 6.43e6T^{2}$$
29 $$1 + 8.23e3iT - 2.05e7T^{2}$$
31 $$1 - 1.98e3T + 2.86e7T^{2}$$
37 $$1 + 9.68e3T + 6.93e7T^{2}$$
41 $$1 + 1.58e4T + 1.15e8T^{2}$$
43 $$1 - 1.87e4T + 1.47e8T^{2}$$
47 $$1 - 5.16e3iT - 2.29e8T^{2}$$
53 $$1 - 5.65e3T + 4.18e8T^{2}$$
59 $$1 + 2.86e4iT - 7.14e8T^{2}$$
61 $$1 - 2.41e3iT - 8.44e8T^{2}$$
67 $$1 - 4.58e4T + 1.35e9T^{2}$$
71 $$1 - 5.53e3T + 1.80e9T^{2}$$
73 $$1 + 5.88e4iT - 2.07e9T^{2}$$
79 $$1 - 7.93e4T + 3.07e9T^{2}$$
83 $$1 + 1.02e5T + 3.93e9T^{2}$$
89 $$1 + 9.52e4T + 5.58e9T^{2}$$
97 $$1 + 5.52e4iT - 8.58e9T^{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

−10.79621536226579908511630843746, −9.788276856460303236631014244157, −8.644919403303540410873445674565, −7.76152460713504150209890178862, −6.70640201619931862408701605543, −5.61334433572892032607885051325, −4.50451190616763123647186633181, −3.72117409411401976800319586549, −1.61311287747234486762581514700, −0.47249389737628958432641094799, 0.791553201967535539842119489294, 2.77015359721821143369826631797, 3.60894845441492569348912247715, 5.31690221730340067553300626069, 6.04665658535531749259083700549, 6.85525267500431021753326590324, 8.469402058146002843996320387089, 8.707431000753820850445919185056, 10.48845161021160637486685847487, 11.06088112613721362298718503511