LMFDB - L-function 2-2800-140.139-c1-0-70

Dirichlet series

L(s) = 1

− 2.96i·3^-s + (2.62 − 0.337i)7^-s − 5.77·9^-s − 3.63i·11^-s − 4.77·13^-s − 4.77·17^-s + 4.57·19^-s + (−1 − 7.77i)21^-s − 5.24·23^-s + 8.20i·27^-s + 4.77·29^-s − 5.92·31^-s − 10.7·33^-s + 11.5i·37^-s + 14.1i·39^-s + ⋯

L(s) = 1

− 1.70i·3^-s + (0.991 − 0.127i)7^-s − 1.92·9^-s − 1.09i·11^-s − 1.32·13^-s − 1.15·17^-s + 1.04·19^-s + (−0.218 − 1.69i)21^-s − 1.09·23^-s + 1.58i·27^-s + 0.886·29^-s − 1.06·31^-s − 1.87·33^-s + 1.89i·37^-s + 2.26i·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2800$ = $2^{4} \cdot 5^{2} \cdot 7$
Sign:	$-0.557 - 0.830i$
Analytic conductor:	$22.3581$
Root analytic conductor:	$4.72843$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2800} (2799, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2800,\ (\ :1/2),\ -0.557 - 0.830i)$

Particular Values

$L(1)$	$\approx$	$0.7631988887$
$L(\frac12)$	$\approx$	$0.7631988887$
$L(\frac{3}{2})$		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 + (-2.62 + 0.337i)T $
good	3	$ 1 + 2.96iT - 3T^{2} $
	11	$ 1 + 3.63iT - 11T^{2} $
	13	$ 1 + 4.77T + 13T^{2} $
	17	$ 1 + 4.77T + 17T^{2} $
	19	$ 1 - 4.57T + 19T^{2} $
	23	$ 1 + 5.24T + 23T^{2} $
	29	$ 1 - 4.77T + 29T^{2} $
	31	$ 1 + 5.92T + 31T^{2} $
	37	$ 1 - 11.5iT - 37T^{2} $
	41	$ 1 + 6iT - 41T^{2} $
	43	$ 1 - 2.02T + 43T^{2} $
	47	$ 1 + 1.61iT - 47T^{2} $
	53	$ 1 + 6iT - 53T^{2} $
	59	$ 1 + 7.27T + 59T^{2} $
	61	$ 1 + 3.54iT - 61T^{2} $
	67	$ 1 + 12.5T + 67T^{2} $
	71	$ 1 - 7.27iT - 71T^{2} $
	73	$ 1 + 6T + 73T^{2} $
	79	$ 1 - 6.85iT - 79T^{2} $
	83	$ 1 + 2.02iT - 83T^{2} $
	89	$ 1 + 12iT - 89T^{2} $
	97	$ 1 + 16.7T + 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

−8.253583806125877959504885714591, −7.45304757777146692379989404550, −7.00656976142707495075817168062, −6.11132013566917365124912546108, −5.39085272594760042619276069720, −4.49406080433575314293404135448, −3.09914767566920531065626294254, −2.22976701335011616969217664752, −1.40275662380499646716648796923, −0.23137893106996043025438834470, 1.96525077264412412264803135261, 2.84691338889371120720421887766, 4.12799742843279904026391514110, 4.54384404940610367993018828646, 5.12954495471429067926495236723, 5.92157310311157716300933492938, 7.25943865323109852946523908165, 7.77757150640559147411922500310, 8.870800561537977453884693341262, 9.349635152808318332924853482507

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7631988887\)
\(L(\frac12)\)	\(\approx\)	\(0.7631988887\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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