LMFDB - L-function 2-2240-28.27-c1-0-24

Dirichlet series

L(s) = 1

− 0.742·3^-s − i·5^-s + (−2.53 − 0.758i)7^-s − 2.44·9^-s + 2.35i·11^-s + 0.960i·13^-s + 0.742i·15^-s + 2.59i·17^-s − 1.82·19^-s + (1.88 + 0.563i)21^-s − 3.24i·23^-s − 25^-s + 4.04·27^-s + 9.30·29^-s − 8.86·31^-s + ⋯

L(s) = 1

− 0.428·3^-s − 0.447i·5^-s + (−0.958 − 0.286i)7^-s − 0.816·9^-s + 0.710i·11^-s + 0.266i·13^-s + 0.191i·15^-s + 0.628i·17^-s − 0.418·19^-s + (0.410 + 0.122i)21^-s − 0.676i·23^-s − 0.200·25^-s + 0.778·27^-s + 1.72·29^-s − 1.59·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$0.958 + 0.286i$
Analytic conductor:	$17.8864$
Root analytic conductor:	$4.22924$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2240} (1791, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2240,\ (\ :1/2),\ 0.958 + 0.286i)$

Particular Values

$L(1)$	$\approx$	$0.9857312621$
$L(\frac12)$	$\approx$	$0.9857312621$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	5	$ 1 + iT $
	7	$ 1 + (2.53 + 0.758i)T $
good	3	$ 1 + 0.742T + 3T^{2} $
	11	$ 1 - 2.35iT - 11T^{2} $
	13	$ 1 - 0.960iT - 13T^{2} $
	17	$ 1 - 2.59iT - 17T^{2} $
	19	$ 1 + 1.82T + 19T^{2} $
	23	$ 1 + 3.24iT - 23T^{2} $
	29	$ 1 - 9.30T + 29T^{2} $
	31	$ 1 + 8.86T + 31T^{2} $
	37	$ 1 - 7.50T + 37T^{2} $
	41	$ 1 + 4.14iT - 41T^{2} $
	43	$ 1 - 9.65iT - 43T^{2} $
	47	$ 1 - 5.52T + 47T^{2} $
	53	$ 1 - 5.21T + 53T^{2} $
	59	$ 1 - 0.391T + 59T^{2} $
	61	$ 1 + 13.8iT - 61T^{2} $
	67	$ 1 - 0.695iT - 67T^{2} $
	71	$ 1 + 16.3iT - 71T^{2} $
	73	$ 1 + 11.7iT - 73T^{2} $
	79	$ 1 + 1.05iT - 79T^{2} $
	83	$ 1 + 1.49T + 83T^{2} $
	89	$ 1 - 9.48iT - 89T^{2} $
	97	$ 1 - 7.40iT - 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

−9.048825102834010213310671973152, −8.306300149864709607546707394399, −7.41111253622638187177972754845, −6.42789370827900235265757599340, −6.07236894552741817443697309043, −4.97155044870564030309409020651, −4.24110532597257306957901314849, −3.20710387675783501679291199393, −2.13408078754714821346513246492, −0.60728166607922759317872667328, 0.66381550175532169604978755764, 2.53247869841470345046740431559, 3.11619562716605982713329254360, 4.11512022884958730277739090259, 5.47276251479315586881550634459, 5.81388429100952729643066715366, 6.67489870439896836385376143550, 7.36125436658468206663786003307, 8.487037344604482574977560658988, 8.967501655271817581638750491935

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9857312621\)
\(L(\frac12)\)	\(\approx\)	\(0.9857312621\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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