LMFDB - L-function 2-84e2-21.20-c1-0-77

Dirichlet series

L(s) = 1

− 1.84·5^-s − 4i·11^-s − 6.62i·13^-s − 2.29·17^-s − 3.06i·19^-s − 5.65i·23^-s − 1.58·25^-s + 9.65i·29^-s − 10.4i·31^-s − 0.242·37^-s + 0.765·41^-s − 9.65·43^-s − 3.06·47^-s − 0.242i·53^-s + 7.39i·55^-s + ⋯

L(s) = 1

− 0.826·5^-s − 1.20i·11^-s − 1.83i·13^-s − 0.556·17^-s − 0.702i·19^-s − 1.17i·23^-s − 0.317·25^-s + 1.79i·29^-s − 1.87i·31^-s − 0.0398·37^-s + 0.119·41^-s − 1.47·43^-s − 0.446·47^-s − 0.0333i·53^-s + 0.996i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7056$ = $2^{4} \cdot 3^{2} \cdot 7^{2}$
Sign:	$-0.860 - 0.508i$
Analytic conductor:	$56.3424$
Root analytic conductor:	$7.50616$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7056} (881, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7056,\ (\ :1/2),\ -0.860 - 0.508i)$

Particular Values

$L(1)$	$\approx$	$0.4955448065$
$L(\frac12)$	$\approx$	$0.4955448065$
$L(\frac{3}{2})$		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 + 1.84T + 5T^{2} $
	11	$ 1 + 4iT - 11T^{2} $
	13	$ 1 + 6.62iT - 13T^{2} $
	17	$ 1 + 2.29T + 17T^{2} $
	19	$ 1 + 3.06iT - 19T^{2} $
	23	$ 1 + 5.65iT - 23T^{2} $
	29	$ 1 - 9.65iT - 29T^{2} $
	31	$ 1 + 10.4iT - 31T^{2} $
	37	$ 1 + 0.242T + 37T^{2} $
	41	$ 1 - 0.765T + 41T^{2} $
	43	$ 1 + 9.65T + 43T^{2} $
	47	$ 1 + 3.06T + 47T^{2} $
	53	$ 1 + 0.242iT - 53T^{2} $
	59	$ 1 - 10.4T + 59T^{2} $
	61	$ 1 - 7.97iT - 61T^{2} $
	67	$ 1 + 4T + 67T^{2} $
	71	$ 1 - 2.34iT - 71T^{2} $
	73	$ 1 + 9.23iT - 73T^{2} $
	79	$ 1 - 5.65T + 79T^{2} $
	83	$ 1 - 11.7T + 83T^{2} $
	89	$ 1 - 3.11T + 89T^{2} $
	97	$ 1 - 9.23iT - 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

−7.67824904316942386906763278151, −6.86882402426926591073657547551, −6.11862938491892457030913641406, −5.40259223599824592421498506207, −4.71057609455543660570303770443, −3.74231644085158090828453020686, −3.18874833787211109190341427992, −2.39523717324682600337041676547, −0.858992068405455961861444396959, −0.14975399604395557624453295986, 1.58006787916933370740563040979, 2.13985210696068990239474342687, 3.43633117588317406167546188717, 4.10816789281039461938455165871, 4.60407975570505139539091480827, 5.43359737441028227307368341438, 6.58567933934321897177849939528, 6.85085338053295584386804694561, 7.67161153018786697541949923582, 8.218451967537947066550550001454

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.4955448065\)
\(L(\frac12)\)	\(\approx\)	\(0.4955448065\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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