LMFDB - L-function 2-70e2-5.4-c1-0-6

Dirichlet series

L(s) = 1

− 2i·3^-s − 9^-s + 2i·13^-s + 6i·17^-s − 4·19^-s − 6i·23^-s − 4i·27^-s − 6·29^-s + 4·31^-s + 2i·37^-s + 4·39^-s − 6·41^-s + 10i·43^-s + 6i·47^-s + 12·51^-s + ⋯

L(s) = 1

− 1.15i·3^-s − 0.333·9^-s + 0.554i·13^-s + 1.45i·17^-s − 0.917·19^-s − 1.25i·23^-s − 0.769i·27^-s − 1.11·29^-s + 0.718·31^-s + 0.328i·37^-s + 0.640·39^-s − 0.937·41^-s + 1.52i·43^-s + 0.875i·47^-s + 1.68·51^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4900$ = $2^{2} \cdot 5^{2} \cdot 7^{2}$
Sign:	$0.447 - 0.894i$
Analytic conductor:	$39.1266$
Root analytic conductor:	$6.25513$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4900} (2549, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4900,\ (\ :1/2),\ 0.447 - 0.894i)$

Particular Values

$L(1)$	$\approx$	$0.9548130672$
$L(\frac12)$	$\approx$	$0.9548130672$
$L(\frac{3}{2})$		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 $
good	3	$ 1 + 2iT - 3T^{2} $
	11	$ 1 + 11T^{2} $
	13	$ 1 - 2iT - 13T^{2} $
	17	$ 1 - 6iT - 17T^{2} $
	19	$ 1 + 4T + 19T^{2} $
	23	$ 1 + 6iT - 23T^{2} $
	29	$ 1 + 6T + 29T^{2} $
	31	$ 1 - 4T + 31T^{2} $
	37	$ 1 - 2iT - 37T^{2} $
	41	$ 1 + 6T + 41T^{2} $
	43	$ 1 - 10iT - 43T^{2} $
	47	$ 1 - 6iT - 47T^{2} $
	53	$ 1 - 6iT - 53T^{2} $
	59	$ 1 - 12T + 59T^{2} $
	61	$ 1 + 2T + 61T^{2} $
	67	$ 1 - 2iT - 67T^{2} $
	71	$ 1 + 12T + 71T^{2} $
	73	$ 1 - 2iT - 73T^{2} $
	79	$ 1 + 8T + 79T^{2} $
	83	$ 1 - 6iT - 83T^{2} $
	89	$ 1 + 6T + 89T^{2} $
	97	$ 1 + 2iT - 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

−8.305789927634818909714806067556, −7.69420788136007504289919218404, −6.82194443513871435098891095537, −6.39468826572413961008538046613, −5.78948010250637930018609365591, −4.55029480576120775637440641325, −4.01524914012158550274887969545, −2.76737731920570098049231575658, −1.94109038548430212508388942562, −1.19144359295722920959211629250, 0.25942250569838280114146347842, 1.80719762122211448153754426983, 2.93992618903816531623346940943, 3.68732340971917116843635770573, 4.36425479567373422894996999906, 5.27392706656227558858189828497, 5.56949570209990711739251984988, 6.83390133838138318916790467467, 7.33744538383109732752886903906, 8.281609211132587024318060712146

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9548130672\)
\(L(\frac12)\)	\(\approx\)	\(0.9548130672\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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