LMFDB - L-function 2-20e2-5.4-c5-0-23

Dirichlet series

L(s) = 1

− 20i·3^-s − 24i·7^-s − 157·9^-s − 124·11^-s + 478i·13^-s + 1.19e3i·17^-s + 3.04e3·19^-s − 480·21^-s − 184i·23^-s − 1.72e3i·27^-s + 3.28e3·29^-s + 5.72e3·31^-s + 2.48e3i·33^-s − 1.03e4i·37^-s + 9.56e3·39^-s + ⋯

L(s) = 1

− 1.28i·3^-s − 0.185i·7^-s − 0.646·9^-s − 0.308·11^-s + 0.784i·13^-s + 1.00i·17^-s + 1.93·19^-s − 0.237·21^-s − 0.0725i·23^-s − 0.454i·27^-s + 0.724·29^-s + 1.07·31^-s + 0.396i·33^-s − 1.24i·37^-s + 1.00·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$400$ = $2^{4} \cdot 5^{2}$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$64.1535$
Root analytic conductor:	$8.00958$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{400} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 400,\ (\ :5/2),\ 0.447 + 0.894i)$

Particular Values

$L(3)$	$\approx$	$2.213731214$
$L(\frac12)$	$\approx$	$2.213731214$
$L(\frac{7}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	5	$ 1 $
good	3	$ 1 + 20iT - 243T^{2} $
	7	$ 1 + 24iT - 1.68e4T^{2} $
	11	$ 1 + 124T + 1.61e5T^{2} $
	13	$ 1 - 478iT - 3.71e5T^{2} $
	17	$ 1 - 1.19e3iT - 1.41e6T^{2} $
	19	$ 1 - 3.04e3T + 2.47e6T^{2} $
	23	$ 1 + 184iT - 6.43e6T^{2} $
	29	$ 1 - 3.28e3T + 2.05e7T^{2} $
	31	$ 1 - 5.72e3T + 2.86e7T^{2} $
	37	$ 1 + 1.03e4iT - 6.93e7T^{2} $
	41	$ 1 + 8.88e3T + 1.15e8T^{2} $
	43	$ 1 - 9.18e3iT - 1.47e8T^{2} $
	47	$ 1 - 2.36e4iT - 2.29e8T^{2} $
	53	$ 1 - 1.16e4iT - 4.18e8T^{2} $
	59	$ 1 - 1.68e4T + 7.14e8T^{2} $
	61	$ 1 + 1.84e4T + 8.44e8T^{2} $
	67	$ 1 + 1.55e4iT - 1.35e9T^{2} $
	71	$ 1 - 3.19e4T + 1.80e9T^{2} $
	73	$ 1 + 4.88e3iT - 2.07e9T^{2} $
	79	$ 1 - 4.45e4T + 3.07e9T^{2} $
	83	$ 1 + 6.73e4iT - 3.93e9T^{2} $
	89	$ 1 + 7.19e4T + 5.58e9T^{2} $
	97	$ 1 + 4.88e4iT - 8.58e9T^{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

−10.32677947444029767226357179652, −9.322895558789729513726829796294, −8.178017451853417018826842341458, −7.47954619801138276879742591691, −6.65169487421542479927565453897, −5.74388682069626053571364493134, −4.39835725666555565380057613268, −2.96820881184214756053887128736, −1.72203002757684432150751786764, −0.804059488792390962004621205534, 0.827159029054178518447462974335, 2.76886489100519995715911051541, 3.60930666756948717608732326822, 4.96529067394087698823290572952, 5.36579602445536263798623328965, 6.88550962116576398699070898360, 7.984927178344406734380933550474, 8.992085778682051071039898587782, 9.947924043345711954527013739433, 10.26726718750677316837038893502

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(3)\)	\(\approx\)	\(2.213731214\)
\(L(\frac12)\)	\(\approx\)	\(2.213731214\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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