LMFDB - L-function 2-800-5.4-c3-0-26

Dirichlet series

L(s) = 1

+ 10.0i·3^-s − 10.5i·7^-s − 74.9·9^-s + 38.9·11^-s − 68.9i·13^-s − 65.9i·17^-s + 49.4·19^-s + 106.·21^-s + 164. i·23^-s − 484. i·27^-s + 170.·29^-s + 166.·31^-s + 392. i·33^-s + 384. i·37^-s + 696.·39^-s + ⋯

L(s) = 1

+ 1.94i·3^-s − 0.571i·7^-s − 2.77·9^-s + 1.06·11^-s − 1.47i·13^-s − 0.940i·17^-s + 0.597·19^-s + 1.11·21^-s + 1.49i·23^-s − 3.45i·27^-s + 1.09·29^-s + 0.964·31^-s + 2.07i·33^-s + 1.70i·37^-s + 2.85·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$800$ = $2^{5} \cdot 5^{2}$
Sign:	$0.447 - 0.894i$
Analytic conductor:	$47.2015$
Root analytic conductor:	$6.87033$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{800} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 800,\ (\ :3/2),\ 0.447 - 0.894i)$

Particular Values

$L(2)$	$\approx$	$2.007287605$
$L(\frac12)$	$\approx$	$2.007287605$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	5	$ 1 $
good	3	$ 1 - 10.0iT - 27T^{2} $
	7	$ 1 + 10.5iT - 343T^{2} $
	11	$ 1 - 38.9T + 1.33e3T^{2} $
	13	$ 1 + 68.9iT - 2.19e3T^{2} $
	17	$ 1 + 65.9iT - 4.91e3T^{2} $
	19	$ 1 - 49.4T + 6.85e3T^{2} $
	23	$ 1 - 164. iT - 1.21e4T^{2} $
	29	$ 1 - 170.T + 2.43e4T^{2} $
	31	$ 1 - 166.T + 2.97e4T^{2} $
	37	$ 1 - 384. iT - 5.06e4T^{2} $
	41	$ 1 + 22.8T + 6.89e4T^{2} $
	43	$ 1 + 136. iT - 7.95e4T^{2} $
	47	$ 1 + 307. iT - 1.03e5T^{2} $
	53	$ 1 - 222iT - 1.48e5T^{2} $
	59	$ 1 + 522.T + 2.05e5T^{2} $
	61	$ 1 - 393.T + 2.26e5T^{2} $
	67	$ 1 + 476. iT - 3.00e5T^{2} $
	71	$ 1 + 4.26T + 3.57e5T^{2} $
	73	$ 1 + 601. iT - 3.89e5T^{2} $
	79	$ 1 - 1.07e3T + 4.93e5T^{2} $
	83	$ 1 + 1.13e3iT - 5.71e5T^{2} $
	89	$ 1 + 479.T + 7.04e5T^{2} $
	97	$ 1 + 635. iT - 9.12e5T^{2} $
show more
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

−10.03984041690687175413450476102, −9.424418920644577817490516700814, −8.588859157896720203248666640886, −7.63754356428033301859803373449, −6.28926913557890733444891186835, −5.27582996643439662958003702839, −4.63949963279984681622712256894, −3.56099518605140875837554258719, −3.01754085758991043333289689593, −0.74814420100568060424759495725, 0.881611301171309135062116763455, 1.83840139639787994774222315789, 2.70839727807963768486179180653, 4.22388890048674477130618800544, 5.70393199827767761766191679272, 6.56157584629239934830272565794, 6.82723794329545857839138839071, 8.039193065144465544654013681838, 8.661733660722410637188322040784, 9.380997476129061957234330870315

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.007287605\)
\(L(\frac12)\)	\(\approx\)	\(2.007287605\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more