LMFDB - L-function 2-2240-5.4-c1-0-44

Dirichlet series

L(s) = 1

− 0.321i·3^-s + (−2.10 + 0.742i)5^-s − i·7^-s + 2.89·9^-s + 4.37·11^-s − 5.86i·13^-s + (0.238 + 0.678i)15^-s + 4.85i·17^-s − 7.75·19^-s − 0.321·21^-s − 1.35i·23^-s + (3.89 − 3.13i)25^-s − 1.89i·27^-s − 0.539·29^-s + 2.97·31^-s + ⋯

L(s) = 1

− 0.185i·3^-s + (−0.943 + 0.332i)5^-s − 0.377i·7^-s + 0.965·9^-s + 1.32·11^-s − 1.62i·13^-s + (0.0616 + 0.175i)15^-s + 1.17i·17^-s − 1.77·19^-s − 0.0701·21^-s − 0.282i·23^-s + (0.779 − 0.626i)25^-s − 0.364i·27^-s − 0.100·29^-s + 0.533·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$0.332 + 0.943i$
Analytic conductor:	$17.8864$
Root analytic conductor:	$4.22924$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2240} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2240,\ (\ :1/2),\ 0.332 + 0.943i)$

Particular Values

$L(1)$	$\approx$	$1.465114594$
$L(\frac12)$	$\approx$	$1.465114594$
$L(\frac{3}{2})$		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 + (2.10 - 0.742i)T $
	7	$ 1 + iT $
good	3	$ 1 + 0.321iT - 3T^{2} $
	11	$ 1 - 4.37T + 11T^{2} $
	13	$ 1 + 5.86iT - 13T^{2} $
	17	$ 1 - 4.85iT - 17T^{2} $
	19	$ 1 + 7.75T + 19T^{2} $
	23	$ 1 + 1.35iT - 23T^{2} $
	29	$ 1 + 0.539T + 29T^{2} $
	31	$ 1 - 2.97T + 31T^{2} $
	37	$ 1 - 6.26iT - 37T^{2} $
	41	$ 1 - 2.64T + 41T^{2} $
	43	$ 1 + 4.64iT - 43T^{2} $
	47	$ 1 + 10.3iT - 47T^{2} $
	53	$ 1 - 0.477iT - 53T^{2} $
	59	$ 1 - 7.75T + 59T^{2} $
	61	$ 1 + 7.57T + 61T^{2} $
	67	$ 1 + 3.79iT - 67T^{2} $
	71	$ 1 - 9.23T + 71T^{2} $
	73	$ 1 - 0.477iT - 73T^{2} $
	79	$ 1 - 1.88T + 79T^{2} $
	83	$ 1 + 15.2iT - 83T^{2} $
	89	$ 1 + 6T + 89T^{2} $
	97	$ 1 + 13.6iT - 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

−8.451498976419150457144127108582, −8.314724761261167874406036377751, −7.25369623979183009850269469117, −6.68199428368724849918548145299, −5.96144058434331793148989215389, −4.55456795228821721050777603301, −4.02235572933946471316617325942, −3.25885984458710767186569355426, −1.83553166227328797718539638765, −0.59156304385804532084819310439, 1.17119617808064601953412422157, 2.30808724706677274581712361477, 3.80795192161127660813587342976, 4.25540321350106664884965938986, 4.88360662930132680278869394981, 6.32500345932592247120896873676, 6.82671387562040942010472874569, 7.57990284718845400759662936915, 8.568073278959866068350931822771, 9.260404950172923991703700960046

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.465114594\)
\(L(\frac12)\)	\(\approx\)	\(1.465114594\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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