LMFDB - L-function 2-2475-5.4-c1-0-11

Dirichlet series

L(s) = 1

− 2.70i·2^-s − 5.34·4^-s − 1.07i·7^-s + 9.04i·8^-s − 11^-s − 4.34i·13^-s − 2.92·14^-s + 13.8·16^-s + 7.75i·17^-s − 5.26·19^-s + 2.70i·22^-s + 2.15i·23^-s − 11.7·26^-s + 5.75i·28^-s + 1.41·29^-s + ⋯

L(s) = 1

− 1.91i·2^-s − 2.67·4^-s − 0.407i·7^-s + 3.19i·8^-s − 0.301·11^-s − 1.20i·13^-s − 0.780·14^-s + 3.45·16^-s + 1.88i·17^-s − 1.20·19^-s + 0.577i·22^-s + 0.449i·23^-s − 2.30·26^-s + 1.08i·28^-s + 0.263·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2475$ = $3^{2} \cdot 5^{2} \cdot 11$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$19.7629$
Root analytic conductor:	$4.44555$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2475} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2475,\ (\ :1/2),\ 0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$0.9070606186$
$L(\frac12)$	$\approx$	$0.9070606186$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
	11	$ 1 + T $
good	2	$ 1 + 2.70iT - 2T^{2} $
	7	$ 1 + 1.07iT - 7T^{2} $
	13	$ 1 + 4.34iT - 13T^{2} $
	17	$ 1 - 7.75iT - 17T^{2} $
	19	$ 1 + 5.26T + 19T^{2} $
	23	$ 1 - 2.15iT - 23T^{2} $
	29	$ 1 - 1.41T + 29T^{2} $
	31	$ 1 + 4.68T + 31T^{2} $
	37	$ 1 - 2iT - 37T^{2} $
	41	$ 1 - 9.41T + 41T^{2} $
	43	$ 1 - 7.60iT - 43T^{2} $
	47	$ 1 - 4.68iT - 47T^{2} $
	53	$ 1 + 0.156iT - 53T^{2} $
	59	$ 1 - 6.15T + 59T^{2} $
	61	$ 1 + 4.15T + 61T^{2} $
	67	$ 1 - 8.68iT - 67T^{2} $
	71	$ 1 - 4.68T + 71T^{2} $
	73	$ 1 + 10.4iT - 73T^{2} $
	79	$ 1 - 8.09T + 79T^{2} $
	83	$ 1 - 11.0iT - 83T^{2} $
	89	$ 1 + 12.8T + 89T^{2} $
	97	$ 1 + 14.6iT - 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

−8.996308824575314681783105746393, −8.273251774845343060782098013221, −7.72269076607254566545608202073, −6.18381353861574135602713721950, −5.40449309743178553647481681833, −4.35883714395087768627411125925, −3.80701178154715864801962924211, −2.91561729531046183238664761078, −1.98880679935300798845728088717, −0.962477082539655104934440808895, 0.38550082024870081196225663818, 2.36960266861791469077501644195, 3.87265958742958798907124254472, 4.62260699065714203854573936568, 5.30892231709228951463812696064, 6.07959538305745469175568880503, 6.89730945643482612790418156988, 7.27800594060438617648647151169, 8.190112826189894619211240229190, 9.033393804666639007121663781654

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9070606186\)
\(L(\frac12)\)	\(\approx\)	\(0.9070606186\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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