LMFDB - L-function 2-585-5.4-c1-0-16

Dirichlet series

L(s) = 1

+ 2.26i·2^-s − 3.10·4^-s + (2.23 + 0.123i)5^-s − 4.96i·7^-s − 2.50i·8^-s + (−0.279 + 5.04i)10^-s + 3.21·11^-s + i·13^-s + 11.2·14^-s − 0.551·16^-s − 0.448i·17^-s + 7.23·19^-s + (−6.94 − 0.383i)20^-s + 7.27i·22^-s + 6.26i·23^-s + ⋯

L(s) = 1

+ 1.59i·2^-s − 1.55·4^-s + (0.998 + 0.0552i)5^-s − 1.87i·7^-s − 0.886i·8^-s + (−0.0882 + 1.59i)10^-s + 0.969·11^-s + 0.277i·13^-s + 3.00·14^-s − 0.137·16^-s − 0.108i·17^-s + 1.65·19^-s + (−1.55 − 0.0858i)20^-s + 1.55i·22^-s + 1.30i·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0552 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0552 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$585$ = $3^{2} \cdot 5 \cdot 13$
Sign:	$0.0552 - 0.998i$
Analytic conductor:	$4.67124$
Root analytic conductor:	$2.16130$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{585} (469, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 585,\ (\ :1/2),\ 0.0552 - 0.998i)$

Particular Values

$L(1)$	$\approx$	$1.25602 + 1.18849i$
$L(\frac12)$	$\approx$	$1.25602 + 1.18849i$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 $
	5	$ 1 + (-2.23 - 0.123i)T $
	13	$ 1 - iT $
good	2	$ 1 - 2.26iT - 2T^{2} $
	7	$ 1 + 4.96iT - 7T^{2} $
	11	$ 1 - 3.21T + 11T^{2} $
	17	$ 1 + 0.448iT - 17T^{2} $
	19	$ 1 - 7.23T + 19T^{2} $
	23	$ 1 - 6.26iT - 23T^{2} $
	29	$ 1 + 2.21T + 29T^{2} $
	31	$ 1 - 2.19T + 31T^{2} $
	37	$ 1 + 8.26iT - 37T^{2} $
	41	$ 1 + 1.79T + 41T^{2} $
	43	$ 1 - 6.21iT - 43T^{2} $
	47	$ 1 - 1.66iT - 47T^{2} $
	53	$ 1 + 1.16iT - 53T^{2} $
	59	$ 1 + 5.88T + 59T^{2} $
	61	$ 1 + 1.76T + 61T^{2} $
	67	$ 1 - 2.73iT - 67T^{2} $
	71	$ 1 + 4.11T + 71T^{2} $
	73	$ 1 + 10.4iT - 73T^{2} $
	79	$ 1 + 1.05T + 79T^{2} $
	83	$ 1 - 5.77iT - 83T^{2} $
	89	$ 1 - 11.1T + 89T^{2} $
	97	$ 1 - 8.37iT - 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

−10.71606302730627415446333974069, −9.577648741408046599980156641282, −9.289775969501770097788236174991, −7.81829465345607930505356157950, −7.24109429860522816998158226310, −6.56735568440847727352086244803, −5.64600087575035032641290654321, −4.65964786540341019336763485052, −3.58180945979055701299161521338, −1.27847229416838481891270252072, 1.41376336064351342738752455795, 2.44905673277558292739683890801, 3.22705968416852356428103774326, 4.78602212180789470879482370809, 5.66892258692812436157855148380, 6.62333554646628487791490811627, 8.446539123021845299779595800497, 9.141680457675033000563027972765, 9.648962438635192824989255652130, 10.46861621618980264919609578614

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.25602 + 1.18849i\)
\(L(\frac12)\)	\(\approx\)	\(1.25602 + 1.18849i\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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