LMFDB - L-function 2-5520-92.91-c1-0-87

Dirichlet series

L(s) = 1

− i·3^-s + i·5^-s + 4.68·7^-s − 9^-s − 3.83·11^-s − 2.63·13^-s + 15^-s − 5.43i·17^-s − 4.36·19^-s − 4.68i·21^-s + (4.47 + 1.71i)23^-s − 25^-s + i·27^-s + 6.33·29^-s − 4.22i·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.447i·5^-s + 1.76·7^-s − 0.333·9^-s − 1.15·11^-s − 0.731·13^-s + 0.258·15^-s − 1.31i·17^-s − 1.00·19^-s − 1.02i·21^-s + (0.933 + 0.357i)23^-s − 0.200·25^-s + 0.192i·27^-s + 1.17·29^-s − 0.758i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5520$ = $2^{4} \cdot 3 \cdot 5 \cdot 23$
Sign:	$-0.776 + 0.630i$
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:	$(2,\ 5520,\ (\ :1/2),\ -0.776 + 0.630i)$

Particular Values

$L(1)$	$\approx$	$1.122823686$
$L(\frac12)$	$\approx$	$1.122823686$
$L(\frac{3}{2})$		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 + (-4.47 - 1.71i)T $
good	7	$ 1 - 4.68T + 7T^{2} $
	11	$ 1 + 3.83T + 11T^{2} $
	13	$ 1 + 2.63T + 13T^{2} $
	17	$ 1 + 5.43iT - 17T^{2} $
	19	$ 1 + 4.36T + 19T^{2} $
	29	$ 1 - 6.33T + 29T^{2} $
	31	$ 1 + 4.22iT - 31T^{2} $
	37	$ 1 + 5.00iT - 37T^{2} $
	41	$ 1 + 5.60T + 41T^{2} $
	43	$ 1 + 12.1T + 43T^{2} $
	47	$ 1 + 5.14iT - 47T^{2} $
	53	$ 1 - 2.61iT - 53T^{2} $
	59	$ 1 + 1.55iT - 59T^{2} $
	61	$ 1 - 11.4iT - 61T^{2} $
	67	$ 1 + 12.6T + 67T^{2} $
	71	$ 1 + 12.0iT - 71T^{2} $
	73	$ 1 + 13.3T + 73T^{2} $
	79	$ 1 + 3.50T + 79T^{2} $
	83	$ 1 - 8.01T + 83T^{2} $
	89	$ 1 + 15.9iT - 89T^{2} $
	97	$ 1 - 10.5iT - 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

−7.62767958818353479433709942378, −7.44666229409330888911088870812, −6.61746711753260522979412249113, −5.56485312570659774398027834791, −4.96384542328276766775334099556, −4.48527155681401063240927338159, −3.04316307676098345828156642201, −2.39500274514435328536986503322, −1.61322940523016569876210061871, −0.27516790986706095877533617957, 1.34032434584163629111185332486, 2.18373350554165936316798908523, 3.15386762181354360973761035308, 4.34037974655395693938743087363, 4.88255392199028417140632622345, 5.14457309577851649392775785664, 6.18035973199302566923891873473, 7.10831135860625236425177408368, 8.102176743979212570967167487216, 8.323989601521538715630647427782

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.122823686\)
\(L(\frac12)\)	\(\approx\)	\(1.122823686\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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