LMFDB - L-function 2-8820-21.20-c1-0-43

Dirichlet series

L(s) = 1

+ 5^-s + 3.53i·11^-s − 0.599i·13^-s − 1.69·17^-s − 7.78i·19^-s − 3.31i·23^-s + 25^-s + 0.456i·29^-s + 1.05i·31^-s − 0.386·37^-s − 0.478·41^-s − 4.21·43^-s + 8.55·47^-s + 3.98i·53^-s + 3.53i·55^-s + ⋯

L(s) = 1

+ 0.447·5^-s + 1.06i·11^-s − 0.166i·13^-s − 0.411·17^-s − 1.78i·19^-s − 0.690i·23^-s + 0.200·25^-s + 0.0846i·29^-s + 0.189i·31^-s − 0.0634·37^-s − 0.0746·41^-s − 0.642·43^-s + 1.24·47^-s + 0.548i·53^-s + 0.476i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$8820$ = $2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2}$
Sign:	$-0.0980 + 0.995i$
Analytic conductor:	$70.4280$
Root analytic conductor:	$8.39214$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{8820} (881, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 8820,\ (\ :1/2),\ -0.0980 + 0.995i)$

Particular Values

$L(1)$	$\approx$	$1.420599185$
$L(\frac12)$	$\approx$	$1.420599185$
$L(\frac{3}{2})$		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 $
	5	$ 1 - T $
	7	$ 1 $
good	11	$ 1 - 3.53iT - 11T^{2} $
	13	$ 1 + 0.599iT - 13T^{2} $
	17	$ 1 + 1.69T + 17T^{2} $
	19	$ 1 + 7.78iT - 19T^{2} $
	23	$ 1 + 3.31iT - 23T^{2} $
	29	$ 1 - 0.456iT - 29T^{2} $
	31	$ 1 - 1.05iT - 31T^{2} $
	37	$ 1 + 0.386T + 37T^{2} $
	41	$ 1 + 0.478T + 41T^{2} $
	43	$ 1 + 4.21T + 43T^{2} $
	47	$ 1 - 8.55T + 47T^{2} $
	53	$ 1 - 3.98iT - 53T^{2} $
	59	$ 1 + 2.60T + 59T^{2} $
	61	$ 1 + 11.0iT - 61T^{2} $
	67	$ 1 - 2.87T + 67T^{2} $
	71	$ 1 + 0.336iT - 71T^{2} $
	73	$ 1 + 3.08iT - 73T^{2} $
	79	$ 1 + 13.6T + 79T^{2} $
	83	$ 1 + 16.2T + 83T^{2} $
	89	$ 1 + 10.6T + 89T^{2} $
	97	$ 1 + 2.18iT - 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

−7.28425761249121261562967029898, −7.03091945187672989797659424834, −6.27753178927883734709831123060, −5.45485728319196776994556708020, −4.70008313602996172660480995564, −4.28850789460348533405934167430, −3.05945196313703571489907908898, −2.42853395668153177389814621405, −1.58285896488390651647377925877, −0.32991635147575727100624257857, 1.08769706850272936544623876378, 1.92766250717594621733930181496, 2.89978090384708892414216546168, 3.68038433251132305266593237813, 4.32910746713529930453544927568, 5.46549556250690458584676191966, 5.76618442229039648167088288208, 6.46667632791737970318336215250, 7.26614335909480946847443871955, 7.985237515439748228566129212412

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.420599185\)
\(L(\frac12)\)	\(\approx\)	\(1.420599185\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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