LMFDB - L-function 2-84e2-28.27-c1-0-20

Dirichlet series

L(s) = 1

− 4.29i·5^-s + 5.06i·11^-s + 3.37i·13^-s − 2.76i·17^-s − 4.70·19^-s − 4.83i·23^-s − 13.4·25^-s − 2.46·29^-s + 5.69·31^-s − 2.33·37^-s + 5.14i·41^-s + 13.0i·43^-s + 5.35·47^-s + 4.22·53^-s + 21.7·55^-s + ⋯

L(s) = 1

− 1.92i·5^-s + 1.52i·11^-s + 0.937i·13^-s − 0.670i·17^-s − 1.07·19^-s − 1.00i·23^-s − 2.69·25^-s − 0.456·29^-s + 1.02·31^-s − 0.383·37^-s + 0.804i·41^-s + 1.98i·43^-s + 0.781·47^-s + 0.580·53^-s + 2.93·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7056$ = $2^{4} \cdot 3^{2} \cdot 7^{2}$
Sign:	$0.933 - 0.358i$
Analytic conductor:	$56.3424$
Root analytic conductor:	$7.50616$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7056} (1567, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7056,\ (\ :1/2),\ 0.933 - 0.358i)$

Particular Values

$L(1)$	$\approx$	$1.452780357$
$L(\frac12)$	$\approx$	$1.452780357$
$L(\frac{3}{2})$		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 $
	3	$ 1 $
	7	$ 1 $
good	5	$ 1 + 4.29iT - 5T^{2} $
	11	$ 1 - 5.06iT - 11T^{2} $
	13	$ 1 - 3.37iT - 13T^{2} $
	17	$ 1 + 2.76iT - 17T^{2} $
	19	$ 1 + 4.70T + 19T^{2} $
	23	$ 1 + 4.83iT - 23T^{2} $
	29	$ 1 + 2.46T + 29T^{2} $
	31	$ 1 - 5.69T + 31T^{2} $
	37	$ 1 + 2.33T + 37T^{2} $
	41	$ 1 - 5.14iT - 41T^{2} $
	43	$ 1 - 13.0iT - 43T^{2} $
	47	$ 1 - 5.35T + 47T^{2} $
	53	$ 1 - 4.22T + 53T^{2} $
	59	$ 1 - 9.60T + 59T^{2} $
	61	$ 1 + 3.87iT - 61T^{2} $
	67	$ 1 - 4.76iT - 67T^{2} $
	71	$ 1 - 12.3iT - 71T^{2} $
	73	$ 1 - 11.5iT - 73T^{2} $
	79	$ 1 + 13.9iT - 79T^{2} $
	83	$ 1 - 7.32T + 83T^{2} $
	89	$ 1 - 14.0iT - 89T^{2} $
	97	$ 1 + 14.5iT - 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

−8.249798310785545038369225823324, −7.28409883808530657975527167135, −6.63680300726412765117423554673, −5.77963195084190104987082397909, −4.85810581990183036446029979672, −4.50943523346333640188541929583, −4.07437786006928115740010443861, −2.49227121512096686214124772919, −1.78610196206107103696302466215, −0.864696493258285896626614822700, 0.42694773948212227045464822455, 2.00768929658618248408535875456, 2.75784731762007943060196487532, 3.56526408146751244005216908629, 3.83983318245568489350030990754, 5.38296748912326996159767740067, 5.91639784875705948190824476445, 6.46053083786914660752944072236, 7.16502957825552481457833858072, 7.84235008167902067439660694185

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.452780357\)
\(L(\frac12)\)	\(\approx\)	\(1.452780357\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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