LMFDB - L-function 2-1008-21.20-c3-0-13

Dirichlet series

L(s) = 1

− 18.5·7^-s − 29.3i·11^-s + 181. i·23^-s − 125·25^-s − 304. i·29^-s − 10.5·37^-s + 534.·43^-s + 343.·49^-s + 768. i·53^-s − 740·67^-s + 1.17e3i·71^-s + 544. i·77^-s + 1.38e3·79^-s − 20.9i·107^-s + 2.27e3·109^-s + ⋯

L(s) = 1

− 0.999·7^-s − 0.805i·11^-s + 1.64i·23^-s − 25^-s − 1.94i·29^-s − 0.0470·37^-s + 1.89·43^-s + 1.00·49^-s + 1.99i·53^-s − 1.34·67^-s + 1.97i·71^-s + 0.805i·77^-s + 1.97·79^-s − 0.0189i·107^-s + 1.99·109^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1008$ = $2^{4} \cdot 3^{2} \cdot 7$
Sign:	$0.577 - 0.816i$
Analytic conductor:	$59.4739$
Root analytic conductor:	$7.71193$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1008} (881, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1008,\ (\ :3/2),\ 0.577 - 0.816i)$

Particular Values

$L(2)$	$\approx$	$1.261932947$
$L(\frac12)$	$\approx$	$1.261932947$
$L(\frac{5}{2})$		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 $
	3	$ 1 $
	7	$ 1 + 18.5T $
good	5	$ 1 + 125T^{2} $
	11	$ 1 + 29.3iT - 1.33e3T^{2} $
	13	$ 1 - 2.19e3T^{2} $
	17	$ 1 + 4.91e3T^{2} $
	19	$ 1 - 6.85e3T^{2} $
	23	$ 1 - 181. iT - 1.21e4T^{2} $
	29	$ 1 + 304. iT - 2.43e4T^{2} $
	31	$ 1 - 2.97e4T^{2} $
	37	$ 1 + 10.5T + 5.06e4T^{2} $
	41	$ 1 + 6.89e4T^{2} $
	43	$ 1 - 534.T + 7.95e4T^{2} $
	47	$ 1 + 1.03e5T^{2} $
	53	$ 1 - 768. iT - 1.48e5T^{2} $
	59	$ 1 + 2.05e5T^{2} $
	61	$ 1 - 2.26e5T^{2} $
	67	$ 1 + 740T + 3.00e5T^{2} $
	71	$ 1 - 1.17e3iT - 3.57e5T^{2} $
	73	$ 1 - 3.89e5T^{2} $
	79	$ 1 - 1.38e3T + 4.93e5T^{2} $
	83	$ 1 + 5.71e5T^{2} $
	89	$ 1 + 7.04e5T^{2} $
	97	$ 1 - 9.12e5T^{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

−9.603212428133466514063835845623, −9.088064204253452117370582049935, −7.943873676624762202920086627845, −7.28534157261273765556800348520, −6.02817261024154411107435213069, −5.77386582626608236300011184105, −4.23938782130990709361146995833, −3.43230304529342847605840724172, −2.38559447445924050929783578172, −0.838009708393414333374423632121, 0.41056246105456451057362411569, 1.97691888578098271903961341539, 3.07218059553523711674112592568, 4.09449751545064842404715346439, 5.08069720115568175951651311484, 6.18211933768406519016636796171, 6.86299330010372017822516675013, 7.69507563290555552287076237445, 8.789947084914737766536826013706, 9.442363247041037842508732239702

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.261932947\)
\(L(\frac12)\)	\(\approx\)	\(1.261932947\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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