LMFDB - L-function 2-1071-17.16-c1-0-21

Dirichlet series

L(s) = 1

− 2.48·2^-s + 4.15·4^-s − 0.675i·5^-s − i·7^-s − 5.35·8^-s + 1.67i·10^-s + 1.80i·11^-s − 2.28·13^-s + 2.48i·14^-s + 4.96·16^-s + (3.67 − 1.86i)17^-s + 0.518·19^-s − 2.80i·20^-s − 4.48i·22^-s − 0.0376i·23^-s + ⋯

L(s) = 1

− 1.75·2^-s + 2.07·4^-s − 0.301i·5^-s − 0.377i·7^-s − 1.89·8^-s + 0.529i·10^-s + 0.544i·11^-s − 0.634·13^-s + 0.663i·14^-s + 1.24·16^-s + (0.891 − 0.453i)17^-s + 0.119·19^-s − 0.627i·20^-s − 0.955i·22^-s − 0.00784i·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1071$ = $3^{2} \cdot 7 \cdot 17$
Sign:	$0.453 + 0.891i$
Analytic conductor:	$8.55197$
Root analytic conductor:	$2.92437$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1071} (883, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1071,\ (\ :1/2),\ 0.453 + 0.891i)$

Particular Values

$L(1)$	$\approx$	$0.6073000953$
$L(\frac12)$	$\approx$	$0.6073000953$
$L(\frac{3}{2})$		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 $
	7	$ 1 + iT $
	17	$ 1 + (-3.67 + 1.86i)T $
good	2	$ 1 + 2.48T + 2T^{2} $
	5	$ 1 + 0.675iT - 5T^{2} $
	11	$ 1 - 1.80iT - 11T^{2} $
	13	$ 1 + 2.28T + 13T^{2} $
	19	$ 1 - 0.518T + 19T^{2} $
	23	$ 1 + 0.0376iT - 23T^{2} $
	29	$ 1 - 0.806iT - 29T^{2} $
	31	$ 1 + 1.28iT - 31T^{2} $
	37	$ 1 + 7.83iT - 37T^{2} $
	41	$ 1 + 0.518iT - 41T^{2} $
	43	$ 1 + 3.57T + 43T^{2} $
	47	$ 1 - 3.98T + 47T^{2} $
	53	$ 1 - 4.09T + 53T^{2} $
	59	$ 1 + 10.2T + 59T^{2} $
	61	$ 1 - 14.5iT - 61T^{2} $
	67	$ 1 - 9.81T + 67T^{2} $
	71	$ 1 + 13.6iT - 71T^{2} $
	73	$ 1 + 12.0iT - 73T^{2} $
	79	$ 1 - 4.99iT - 79T^{2} $
	83	$ 1 + 2.26T + 83T^{2} $
	89	$ 1 - 13.2T + 89T^{2} $
	97	$ 1 + 13.3iT - 97T^{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

−9.597506720777925485758728210451, −9.086437559395529114978734326048, −8.148707236836684585549153990715, −7.40137219728201846601116988458, −6.92968453989699884282277763499, −5.68333614737282175406749872619, −4.52887475631487771864218428402, −3.00920659873667225451339598046, −1.81840686931564249675152172910, −0.59166988843703493181867154096, 1.05974093296693408061635605290, 2.36147605405511391774619153971, 3.34677680768542383581610109637, 5.06110906545939861727327173839, 6.19568153793027331370221835231, 6.93036236889963939156429236468, 7.83540492110659555230987774330, 8.410827283149381737142875689112, 9.212364636212692331274517868791, 9.967483360092499671097294864222

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

\(L(1)\)	\(\approx\)	\(0.6073000953\)
\(L(\frac12)\)	\(\approx\)	\(0.6073000953\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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