LMFDB - L-function 2-8820-21.20-c1-0-40

Dirichlet series

L(s) = 1

− 5^-s − 1.79i·11^-s + 2.19i·13^-s + 0.936·17^-s − 0.422i·19^-s − 4.08i·23^-s + 25^-s + 8.43i·29^-s − 10.6i·31^-s + 8.37·37^-s + 6.29·41^-s − 9.56·43^-s − 3.04·47^-s + 6.64i·53^-s + 1.79i·55^-s + ⋯

L(s) = 1

− 0.447·5^-s − 0.539i·11^-s + 0.609i·13^-s + 0.227·17^-s − 0.0970i·19^-s − 0.852i·23^-s + 0.200·25^-s + 1.56i·29^-s − 1.90i·31^-s + 1.37·37^-s + 0.982·41^-s − 1.45·43^-s − 0.444·47^-s + 0.912i·53^-s + 0.241i·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

$L(1)$	$\approx$	$1.292897978$
$L(\frac12)$	$\approx$	$1.292897978$
$L(\frac{3}{2})$		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 + 1.79iT - 11T^{2} $
	13	$ 1 - 2.19iT - 13T^{2} $
	17	$ 1 - 0.936T + 17T^{2} $
	19	$ 1 + 0.422iT - 19T^{2} $
	23	$ 1 + 4.08iT - 23T^{2} $
	29	$ 1 - 8.43iT - 29T^{2} $
	31	$ 1 + 10.6iT - 31T^{2} $
	37	$ 1 - 8.37T + 37T^{2} $
	41	$ 1 - 6.29T + 41T^{2} $
	43	$ 1 + 9.56T + 43T^{2} $
	47	$ 1 + 3.04T + 47T^{2} $
	53	$ 1 - 6.64iT - 53T^{2} $
	59	$ 1 + 12.7T + 59T^{2} $
	61	$ 1 + 1.94iT - 61T^{2} $
	67	$ 1 - 7.16T + 67T^{2} $
	71	$ 1 - 4.91iT - 71T^{2} $
	73	$ 1 + 4.61iT - 73T^{2} $
	79	$ 1 - 5.56T + 79T^{2} $
	83	$ 1 + 15.1T + 83T^{2} $
	89	$ 1 - 4.30T + 89T^{2} $
	97	$ 1 - 4.55iT - 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

−7.71272454756011466736762099063, −6.87350813846478088470694313165, −6.27776470693028792675774138911, −5.58328790012585768646510514698, −4.66335332601386152447024342905, −4.14436604180653175723968170324, −3.26582481330834730390987495614, −2.53651483560136201992203538600, −1.45121254565525005911078549570, −0.35712560369384037939295901877, 0.917523505030216052924157012771, 1.94503844544670169870148071365, 2.95499761451088895971433587167, 3.59356887469179475580816192930, 4.46911265711764520370194645423, 5.06858181621698766217530337554, 5.87738280258739940786476918422, 6.56643469812268930115013870169, 7.34983590475815503020884304127, 7.88835990455840714161892415663

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.292897978\)
\(L(\frac12)\)	\(\approx\)	\(1.292897978\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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