LMFDB - L-function 2-15e2-5.4-c5-0-16

Dirichlet series

L(s) = 1

+ 32·4^-s − 25i·7^-s + 775i·13^-s + 1.02e3·16^-s + 1.71e3·19^-s − 800i·28^-s + 2.72e3·31^-s + 1.65e4i·37^-s − 2.24e4i·43^-s + 1.61e4·49^-s + 2.48e4i·52^-s + 5.69e4·61^-s + 3.27e4·64^-s + 7.34e4i·67^-s + 1.45e3i·73^-s + ⋯

L(s) = 1

+ 4^-s − 0.192i·7^-s + 1.27i·13^-s + 16^-s + 1.08·19^-s − 0.192i·28^-s + 0.508·31^-s + 1.98i·37^-s − 1.85i·43^-s + 0.962·49^-s + 1.27i·52^-s + 1.95·61^-s + 64^-s + 1.99i·67^-s + 0.0318i·73^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$225$ = $3^{2} \cdot 5^{2}$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$36.0863$
Root analytic conductor:	$6.00719$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{225} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 225,\ (\ :5/2),\ 0.894 - 0.447i)$

Particular Values

$L(3)$	$\approx$	$2.697418892$
$L(\frac12)$	$\approx$	$2.697418892$
$L(\frac{7}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 $
	5	$ 1 $
good	2	$ 1 - 32T^{2} $
	7	$ 1 + 25iT - 1.68e4T^{2} $
	11	$ 1 + 1.61e5T^{2} $
	13	$ 1 - 775iT - 3.71e5T^{2} $
	17	$ 1 - 1.41e6T^{2} $
	19	$ 1 - 1.71e3T + 2.47e6T^{2} $
	23	$ 1 - 6.43e6T^{2} $
	29	$ 1 + 2.05e7T^{2} $
	31	$ 1 - 2.72e3T + 2.86e7T^{2} $
	37	$ 1 - 1.65e4iT - 6.93e7T^{2} $
	41	$ 1 + 1.15e8T^{2} $
	43	$ 1 + 2.24e4iT - 1.47e8T^{2} $
	47	$ 1 - 2.29e8T^{2} $
	53	$ 1 - 4.18e8T^{2} $
	59	$ 1 + 7.14e8T^{2} $
	61	$ 1 - 5.69e4T + 8.44e8T^{2} $
	67	$ 1 - 7.34e4iT - 1.35e9T^{2} $
	71	$ 1 + 1.80e9T^{2} $
	73	$ 1 - 1.45e3iT - 2.07e9T^{2} $
	79	$ 1 - 1.00e5T + 3.07e9T^{2} $
	83	$ 1 - 3.93e9T^{2} $
	89	$ 1 + 5.58e9T^{2} $
	97	$ 1 - 1.77e5iT - 8.58e9T^{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

−11.65807709002491445058933728566, −10.53388525137663684489134688356, −9.637708569281348109308726435558, −8.400496092143266394377790076234, −7.22906019182699561313754904596, −6.55609931903575997889018931677, −5.28653744020645626499604979846, −3.81524134887619496324232675041, −2.47850559886211993069216951009, −1.20688678904304418736697766709, 0.895406923448672246553255425026, 2.41475497133123208653350340928, 3.47440328665539204454983796001, 5.24725017484214509276782126193, 6.15057811390926630581463099327, 7.35508522095215685753854071448, 8.092792667407989877706673524119, 9.482852481736386759860752571637, 10.47579246510730645472078592014, 11.27275064373820535899655918326

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 $\F_{q}$
Belyi maps

\(L(3)\)	\(\approx\)	\(2.697418892\)
\(L(\frac12)\)	\(\approx\)	\(2.697418892\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more