LMFDB - L-function 2-2800-5.4-c1-0-51

Dirichlet series

L(s) = 1

− 1.56i·3^-s − i·7^-s + 0.561·9^-s − 1.56·11^-s − 6.68i·13^-s + 7.56i·17^-s − 7.12·19^-s − 1.56·21^-s + 3.12i·23^-s − 5.56i·27^-s − 0.438·29^-s − 6.24·31^-s + 2.43i·33^-s − 8.24i·37^-s − 10.4·39^-s + ⋯

L(s) = 1

− 0.901i·3^-s − 0.377i·7^-s + 0.187·9^-s − 0.470·11^-s − 1.85i·13^-s + 1.83i·17^-s − 1.63·19^-s − 0.340·21^-s + 0.651i·23^-s − 1.07i·27^-s − 0.0814·29^-s − 1.12·31^-s + 0.424i·33^-s − 1.35i·37^-s − 1.67·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2800$ = $2^{4} \cdot 5^{2} \cdot 7$
Sign:	$-0.894 - 0.447i$
Analytic conductor:	$22.3581$
Root analytic conductor:	$4.72843$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2800} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2800,\ (\ :1/2),\ -0.894 - 0.447i)$

Particular Values

$L(1)$	$\approx$	$0.5562907995$
$L(\frac12)$	$\approx$	$0.5562907995$
$L(\frac{3}{2})$		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 $
	7	$ 1 + iT $
good	3	$ 1 + 1.56iT - 3T^{2} $
	11	$ 1 + 1.56T + 11T^{2} $
	13	$ 1 + 6.68iT - 13T^{2} $
	17	$ 1 - 7.56iT - 17T^{2} $
	19	$ 1 + 7.12T + 19T^{2} $
	23	$ 1 - 3.12iT - 23T^{2} $
	29	$ 1 + 0.438T + 29T^{2} $
	31	$ 1 + 6.24T + 31T^{2} $
	37	$ 1 + 8.24iT - 37T^{2} $
	41	$ 1 + 1.12T + 41T^{2} $
	43	$ 1 + 7.12iT - 43T^{2} $
	47	$ 1 + 2.43iT - 47T^{2} $
	53	$ 1 - 13.1iT - 53T^{2} $
	59	$ 1 + 4T + 59T^{2} $
	61	$ 1 + 6.87T + 61T^{2} $
	67	$ 1 + 2.24iT - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 - 4.24iT - 73T^{2} $
	79	$ 1 - 0.684T + 79T^{2} $
	83	$ 1 - 12iT - 83T^{2} $
	89	$ 1 + 5.12T + 89T^{2} $
	97	$ 1 - 1.31iT - 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

−8.139702177186718295064851056980, −7.68125393183190333371052979456, −6.97563581251376831187189743570, −6.00528596244521783657316911190, −5.57682370952244326304818915735, −4.28508356916486569571280198773, −3.54492701248456895948612311908, −2.33560595878601600038479086338, −1.46692124714175298088941314771, −0.16885591377321239388541542978, 1.77410993889226884529692869285, 2.72375619619036583174805094324, 3.82989121372762738932849321672, 4.69953180893672835199418891330, 4.95881985394922716778700282093, 6.30624799851939722179968751244, 6.83984378653936473633389449806, 7.73704539490718563216177248977, 8.825971784633041005245717511821, 9.179059114298190855554102767937

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.5562907995\)
\(L(\frac12)\)	\(\approx\)	\(0.5562907995\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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