Properties

 Degree 2 Conductor $2^{5} \cdot 3^{2} \cdot 5$ Sign $-0.423 - 0.905i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (0.254 + 2.22i)5-s + 2.64i·7-s + 1.51i·11-s + 3.87·13-s + 3.31i·17-s − 7.08i·19-s + 4.82i·23-s + (−4.87 + 1.12i)25-s + 2.18i·29-s + 7.36·31-s + (−5.87 + 0.671i)35-s − 7.87·37-s − 8.72·41-s − 1.01·43-s + 7.08i·47-s + ⋯
 L(s)  = 1 + (0.113 + 0.993i)5-s + 0.998i·7-s + 0.456i·11-s + 1.07·13-s + 0.803i·17-s − 1.62i·19-s + 1.00i·23-s + (−0.974 + 0.225i)25-s + 0.405i·29-s + 1.32·31-s + (−0.992 + 0.113i)35-s − 1.29·37-s − 1.36·41-s − 0.155·43-s + 1.03i·47-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.423 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$1440$$    =    $$2^{5} \cdot 3^{2} \cdot 5$$ $$\varepsilon$$ = $-0.423 - 0.905i$ motivic weight = $$1$$ character : $\chi_{1440} (1009, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 1440,\ (\ :1/2),\ -0.423 - 0.905i)$$ $$L(1)$$ $$\approx$$ $$1.535459161$$ $$L(\frac12)$$ $$\approx$$ $$1.535459161$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1 + (-0.254 - 2.22i)T$$
good7 $$1 - 2.64iT - 7T^{2}$$
11 $$1 - 1.51iT - 11T^{2}$$
13 $$1 - 3.87T + 13T^{2}$$
17 $$1 - 3.31iT - 17T^{2}$$
19 $$1 + 7.08iT - 19T^{2}$$
23 $$1 - 4.82iT - 23T^{2}$$
29 $$1 - 2.18iT - 29T^{2}$$
31 $$1 - 7.36T + 31T^{2}$$
37 $$1 + 7.87T + 37T^{2}$$
41 $$1 + 8.72T + 41T^{2}$$
43 $$1 + 1.01T + 43T^{2}$$
47 $$1 - 7.08iT - 47T^{2}$$
53 $$1 + 4.50T + 53T^{2}$$
59 $$1 + 6.79iT - 59T^{2}$$
61 $$1 - 3.60iT - 61T^{2}$$
67 $$1 - 1.01T + 67T^{2}$$
71 $$1 + 6.72T + 71T^{2}$$
73 $$1 - 15.5iT - 73T^{2}$$
79 $$1 + 7.36T + 79T^{2}$$
83 $$1 - 7.74T + 83T^{2}$$
89 $$1 - 14.7T + 89T^{2}$$
97 $$1 + 11.1iT - 97T^{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

−9.810226818428075261460677532192, −8.930991393144291870459046366458, −8.314328801644740341250531006040, −7.22120526023901260487492168697, −6.51786876688558097952813339302, −5.80530347700276820238898848802, −4.83587466011782058375013325307, −3.58491394484793003804569950634, −2.77057768298575599662989351449, −1.68848372232822419363629281888, 0.63387276770046806625883227740, 1.69220754793623285507704848024, 3.35456174967073238826948458896, 4.14661494412211771573830890763, 5.02677295304508982351845960538, 5.98571442105713457459918030359, 6.75786698426310170069891003407, 7.911177608819110983359297392566, 8.393844923273724910564295440307, 9.180513667354531415699426001228