LMFDB - L-function 2-2800-5.4-c1-0-47

Dirichlet series

L(s) = 1

− 1.56i·3^-s − i·7^-s + 0.561·9^-s + 1.56·11^-s + 0.438i·13^-s + 0.438i·17^-s − 7.12·19^-s − 1.56·21^-s − 3.12i·23^-s − 5.56i·27^-s − 6.68·29^-s − 2.43i·33^-s − 6i·37^-s + 0.684·39^-s + 5.12·41^-s + ⋯

L(s) = 1

− 0.901i·3^-s − 0.377i·7^-s + 0.187·9^-s + 0.470·11^-s + 0.121i·13^-s + 0.106i·17^-s − 1.63·19^-s − 0.340·21^-s − 0.651i·23^-s − 1.07i·27^-s − 1.24·29^-s − 0.424i·33^-s − 0.986i·37^-s + 0.109·39^-s + 0.800·41^-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

$L(1)$	$\approx$	$1.263521550$
$L(\frac12)$	$\approx$	$1.263521550$
$L(\frac{3}{2})$		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 - 0.438iT - 13T^{2} $
	17	$ 1 - 0.438iT - 17T^{2} $
	19	$ 1 + 7.12T + 19T^{2} $
	23	$ 1 + 3.12iT - 23T^{2} $
	29	$ 1 + 6.68T + 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 6iT - 37T^{2} $
	41	$ 1 - 5.12T + 41T^{2} $
	43	$ 1 + 0.876iT - 43T^{2} $
	47	$ 1 + 8.68iT - 47T^{2} $
	53	$ 1 + 5.12iT - 53T^{2} $
	59	$ 1 + 4T + 59T^{2} $
	61	$ 1 - 15.3T + 61T^{2} $
	67	$ 1 - 10.2iT - 67T^{2} $
	71	$ 1 + 8T + 71T^{2} $
	73	$ 1 + 12.2iT - 73T^{2} $
	79	$ 1 + 2.43T + 79T^{2} $
	83	$ 1 + 4iT - 83T^{2} $
	89	$ 1 - 1.12T + 89T^{2} $
	97	$ 1 + 5.80iT - 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

−8.417970029565876307586477310058, −7.61480079074203643030036007921, −6.93192376603501862357408550291, −6.42087145397753030102590998480, −5.55622158556878941538398932387, −4.34551266295589190708727160942, −3.82919342115948216994089682906, −2.38940677915063738726660627051, −1.66556931333846498129795453625, −0.39792314554983039333910740301, 1.48428517972698366486251587448, 2.64221845860577361641568807484, 3.75713174986157639178994331638, 4.30388525667918335338218214118, 5.16225057799263856675908291233, 5.99801781294293740747761391398, 6.77894734661240783235081473103, 7.68534557230671574830574214640, 8.523851688002428944185853336417, 9.281815939424230875719285939011

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.263521550\)
\(L(\frac12)\)	\(\approx\)	\(1.263521550\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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