LMFDB - L-function 2-2240-5.4-c1-0-16

Dirichlet series

L(s) = 1

− 2.85i·3^-s + (0.649 + 2.13i)5^-s − i·7^-s − 5.15·9^-s − 4.61·11^-s + 0.332i·13^-s + (6.11 − 1.85i)15^-s + 7.60i·17^-s + 1.28·19^-s − 2.85·21^-s + 3.71i·23^-s + (−4.15 + 2.78i)25^-s + 6.15i·27^-s + 2.44·29^-s + 8.55·31^-s + ⋯

L(s) = 1

− 1.64i·3^-s + (0.290 + 0.956i)5^-s − 0.377i·7^-s − 1.71·9^-s − 1.39·11^-s + 0.0922i·13^-s + (1.57 − 0.479i)15^-s + 1.84i·17^-s + 0.294·19^-s − 0.623·21^-s + 0.773i·23^-s + (−0.831 + 0.556i)25^-s + 1.18i·27^-s + 0.453·29^-s + 1.53·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$0.956 - 0.290i$
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:	$(2,\ 2240,\ (\ :1/2),\ 0.956 - 0.290i)$

Particular Values

$L(1)$	$\approx$	$1.313983158$
$L(\frac12)$	$\approx$	$1.313983158$
$L(\frac{3}{2})$		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 + (-0.649 - 2.13i)T $
	7	$ 1 + iT $
good	3	$ 1 + 2.85iT - 3T^{2} $
	11	$ 1 + 4.61T + 11T^{2} $
	13	$ 1 - 0.332iT - 13T^{2} $
	17	$ 1 - 7.60iT - 17T^{2} $
	19	$ 1 - 1.28T + 19T^{2} $
	23	$ 1 - 3.71iT - 23T^{2} $
	29	$ 1 - 2.44T + 29T^{2} $
	31	$ 1 - 8.55T + 31T^{2} $
	37	$ 1 + 5.56iT - 37T^{2} $
	41	$ 1 - 7.71T + 41T^{2} $
	43	$ 1 + 9.71iT - 43T^{2} $
	47	$ 1 - 8.75iT - 47T^{2} $
	53	$ 1 - 12.2iT - 53T^{2} $
	59	$ 1 + 1.28T + 59T^{2} $
	61	$ 1 - 3.01T + 61T^{2} $
	67	$ 1 - 12.3iT - 67T^{2} $
	71	$ 1 - 2.99T + 71T^{2} $
	73	$ 1 - 12.2iT - 73T^{2} $
	79	$ 1 + 0.949T + 79T^{2} $
	83	$ 1 + 3.81iT - 83T^{2} $
	89	$ 1 + 6T + 89T^{2} $
	97	$ 1 - 1.61iT - 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

−8.806502471931496864736108000543, −7.87304939685750426098970308912, −7.62368805741176328207483343092, −6.81509018666067044363644470861, −6.06736297029573808771950834312, −5.56545260162574663232593112498, −4.05791063557161425005916621378, −2.86630510703429107910722008103, −2.23592569627127486867300547767, −1.14875821880919529245146554179, 0.49037125656644304994425319159, 2.49394194373985360154089748798, 3.16129224065109864639264292505, 4.52982104177109828257180288849, 4.90016906235959747250890268637, 5.40171942587509889870314213754, 6.41765106601821610546514016820, 7.80034280377303672633399057973, 8.427974120231622128158178087299, 9.144852955942263737360185229758

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.313983158\)
\(L(\frac12)\)	\(\approx\)	\(1.313983158\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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