LMFDB - L-function 2-5520-92.91-c1-0-52

Dirichlet series

L(s) = 1

− i·3^-s + i·5^-s + 4.84·7^-s − 9^-s + 3.09·11^-s − 4.53·13^-s + 15^-s + 1.04i·17^-s + 3.05·19^-s − 4.84i·21^-s + (4.79 + 0.0971i)23^-s − 25^-s + i·27^-s − 2.44·29^-s + 3.08i·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.447i·5^-s + 1.83·7^-s − 0.333·9^-s + 0.933·11^-s − 1.25·13^-s + 0.258·15^-s + 0.254i·17^-s + 0.701·19^-s − 1.05i·21^-s + (0.999 + 0.0202i)23^-s − 0.200·25^-s + 0.192i·27^-s − 0.454·29^-s + 0.553i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0202i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5520$ = $2^{4} \cdot 3 \cdot 5 \cdot 23$
Sign:	$0.999 + 0.0202i$
Analytic conductor:	$44.0774$
Root analytic conductor:	$6.63908$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5520} (1471, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5520,\ (\ :1/2),\ 0.999 + 0.0202i)$

Particular Values

$L(1)$	$\approx$	$2.650143274$
$L(\frac12)$	$\approx$	$2.650143274$
$L(\frac{3}{2})$		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 + iT $
	5	$ 1 - iT $
	23	$ 1 + (-4.79 - 0.0971i)T $
good	7	$ 1 - 4.84T + 7T^{2} $
	11	$ 1 - 3.09T + 11T^{2} $
	13	$ 1 + 4.53T + 13T^{2} $
	17	$ 1 - 1.04iT - 17T^{2} $
	19	$ 1 - 3.05T + 19T^{2} $
	29	$ 1 + 2.44T + 29T^{2} $
	31	$ 1 - 3.08iT - 31T^{2} $
	37	$ 1 - 3.78iT - 37T^{2} $
	41	$ 1 - 4.04T + 41T^{2} $
	43	$ 1 - 2.53T + 43T^{2} $
	47	$ 1 + 3.83iT - 47T^{2} $
	53	$ 1 - 8.14iT - 53T^{2} $
	59	$ 1 - 4.64iT - 59T^{2} $
	61	$ 1 + 11.3iT - 61T^{2} $
	67	$ 1 - 3.77T + 67T^{2} $
	71	$ 1 + 13.2iT - 71T^{2} $
	73	$ 1 - 2.53T + 73T^{2} $
	79	$ 1 - 1.04T + 79T^{2} $
	83	$ 1 + 4.24T + 83T^{2} $
	89	$ 1 - 3.89iT - 89T^{2} $
	97	$ 1 + 5.72iT - 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

−7.923724183995082551897631741188, −7.48589197377165484529494259812, −6.93349899488476770314085622779, −6.04836516112801190580600194474, −5.12002949659959517528988368617, −4.70876991475755596324253585988, −3.66356730454853034176289733873, −2.61585734854458760936061117551, −1.80363997303794856672112572002, −1.01168468848430923748269190015, 0.855251217181061686728592280293, 1.82820872055293867223018615502, 2.75597185534434651999821651948, 4.00271766175966704242198957363, 4.50929241208421410222531756368, 5.21076955086117476672698831272, 5.62727862731267290650307035817, 6.92840228900874157262142488996, 7.53419064274351981000497546627, 8.160080707283992202933923098848

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.650143274\)
\(L(\frac12)\)	\(\approx\)	\(2.650143274\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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