LMFDB - L-function 2-42e2-21.20-c3-0-15

Dirichlet series

L(s) = 1

− 8.54·5^-s − 31.7i·11^-s + 9.92i·13^-s − 127.·17^-s + 116. i·19^-s − 64.4i·23^-s − 52.0·25^-s + 113. i·29^-s + 7.31i·31^-s + 369.·37^-s + 211.·41^-s − 432.·43^-s + 400.·47^-s − 140. i·53^-s + 270. i·55^-s + ⋯

L(s) = 1

− 0.764·5^-s − 0.869i·11^-s + 0.211i·13^-s − 1.81·17^-s + 1.40i·19^-s − 0.584i·23^-s − 0.416·25^-s + 0.723i·29^-s + 0.0423i·31^-s + 1.64·37^-s + 0.807·41^-s − 1.53·43^-s + 1.24·47^-s − 0.364i·53^-s + 0.664i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1764$ = $2^{2} \cdot 3^{2} \cdot 7^{2}$
Sign:	$0.970 + 0.239i$
Analytic conductor:	$104.079$
Root analytic conductor:	$10.2019$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1764} (881, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1764,\ (\ :3/2),\ 0.970 + 0.239i)$

Particular Values

$L(2)$	$\approx$	$1.166243041$
$L(\frac12)$	$\approx$	$1.166243041$
$L(\frac{5}{2})$		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 $
	3	$ 1 $
	7	$ 1 $
good	5	$ 1 + 8.54T + 125T^{2} $
	11	$ 1 + 31.7iT - 1.33e3T^{2} $
	13	$ 1 - 9.92iT - 2.19e3T^{2} $
	17	$ 1 + 127.T + 4.91e3T^{2} $
	19	$ 1 - 116. iT - 6.85e3T^{2} $
	23	$ 1 + 64.4iT - 1.21e4T^{2} $
	29	$ 1 - 113. iT - 2.43e4T^{2} $
	31	$ 1 - 7.31iT - 2.97e4T^{2} $
	37	$ 1 - 369.T + 5.06e4T^{2} $
	41	$ 1 - 211.T + 6.89e4T^{2} $
	43	$ 1 + 432.T + 7.95e4T^{2} $
	47	$ 1 - 400.T + 1.03e5T^{2} $
	53	$ 1 + 140. iT - 1.48e5T^{2} $
	59	$ 1 + 518.T + 2.05e5T^{2} $
	61	$ 1 + 27.2iT - 2.26e5T^{2} $
	67	$ 1 + 136.T + 3.00e5T^{2} $
	71	$ 1 + 604. iT - 3.57e5T^{2} $
	73	$ 1 + 48.4iT - 3.89e5T^{2} $
	79	$ 1 + 831.T + 4.93e5T^{2} $
	83	$ 1 + 37.2T + 5.71e5T^{2} $
	89	$ 1 + 470.T + 7.04e5T^{2} $
	97	$ 1 - 522. iT - 9.12e5T^{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

−8.747584802727293437345373956142, −8.200094018422530990260030544477, −7.40889349850282337164631166448, −6.46262822383273003163242407143, −5.81854987950136528172386414465, −4.58717478088768909660615245593, −3.97305587648347161040206488972, −3.00002791340872085469114180309, −1.83349270654882792014818732261, −0.46285627123880164551955543547, 0.53067611549071209338045392124, 2.02870459964862530369694778242, 2.91783840824593258582977695336, 4.29957799955049697331494790019, 4.49499194829579967998705942643, 5.76780926794084240149957925500, 6.75642297693214194289901678473, 7.35870091050898855250858580277, 8.119046858521814380968139400039, 9.020291531643046569241535085756

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.166243041\)
\(L(\frac12)\)	\(\approx\)	\(1.166243041\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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