LMFDB - L-function 2-84e2-21.20-c1-0-27

Dirichlet series

L(s) = 1

− 0.765·5^-s − 2i·11^-s + 0.317i·13^-s − 5.54·17^-s + 3.69i·19^-s − 3.17i·23^-s − 4.41·25^-s − 6.82i·29^-s + 6.75i·31^-s − 0.242·37^-s + 2.74·41^-s − 6.82·43^-s + 11.9·47^-s + 12.2i·53^-s + 1.53i·55^-s + ⋯

L(s) = 1

− 0.342·5^-s − 0.603i·11^-s + 0.0879i·13^-s − 1.34·17^-s + 0.847i·19^-s − 0.661i·23^-s − 0.882·25^-s − 1.26i·29^-s + 1.21i·31^-s − 0.0398·37^-s + 0.428·41^-s − 1.04·43^-s + 1.74·47^-s + 1.68i·53^-s + 0.206i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7056$ = $2^{4} \cdot 3^{2} \cdot 7^{2}$
Sign:	$0.896 - 0.442i$
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:	$(2,\ 7056,\ (\ :1/2),\ 0.896 - 0.442i)$

Particular Values

$L(1)$	$\approx$	$1.363370023$
$L(\frac12)$	$\approx$	$1.363370023$
$L(\frac{3}{2})$		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 + 0.765T + 5T^{2} $
	11	$ 1 + 2iT - 11T^{2} $
	13	$ 1 - 0.317iT - 13T^{2} $
	17	$ 1 + 5.54T + 17T^{2} $
	19	$ 1 - 3.69iT - 19T^{2} $
	23	$ 1 + 3.17iT - 23T^{2} $
	29	$ 1 + 6.82iT - 29T^{2} $
	31	$ 1 - 6.75iT - 31T^{2} $
	37	$ 1 + 0.242T + 37T^{2} $
	41	$ 1 - 2.74T + 41T^{2} $
	43	$ 1 + 6.82T + 43T^{2} $
	47	$ 1 - 11.9T + 47T^{2} $
	53	$ 1 - 12.2iT - 53T^{2} $
	59	$ 1 - 13.2T + 59T^{2} $
	61	$ 1 + 3.56iT - 61T^{2} $
	67	$ 1 - 4.48T + 67T^{2} $
	71	$ 1 - 9.31iT - 71T^{2} $
	73	$ 1 + 11.8iT - 73T^{2} $
	79	$ 1 - 11.3T + 79T^{2} $
	83	$ 1 - 4.32T + 83T^{2} $
	89	$ 1 + 1.66T + 89T^{2} $
	97	$ 1 - 11.8iT - 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

−8.094428789661900243946808959720, −7.30966217723995859459736373348, −6.54873842509669207651698364282, −5.98044781242516734999047483750, −5.17751901710621948508787162775, −4.23873796260130800583157286124, −3.80466248914284176579091936725, −2.73788075551882949494270885236, −1.95530772859399480074207728494, −0.68721712199752029433449252416, 0.48726836870764441151489964093, 1.86252699715047978992491251418, 2.54610498456801539280325423888, 3.65079408581319162283163072112, 4.24569896198141240346188516900, 5.03732980912410758310580039322, 5.71120362870700763611733309687, 6.74506522355782640884002251949, 7.06327574467328024582037198964, 7.87799284731825052248440663862

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.363370023\)
\(L(\frac12)\)	\(\approx\)	\(1.363370023\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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