LMFDB - L-function 2-1440-5.4-c3-0-5

Dirichlet series

L(s) = 1

+ (0.582 + 11.1i)5^-s − 22.1i·7^-s − 27.1·11^-s + 70.3i·13^-s + 73.3i·17^-s + 110.·19^-s − 107. i·23^-s + (−124. + 13.0i)25^-s + 68.6·29^-s − 137.·31^-s + (247. − 12.8i)35^-s + 60.3i·37^-s − 95.1·41^-s − 501. i·43^-s + 439. i·47^-s + ⋯

L(s) = 1

+ (0.0521 + 0.998i)5^-s − 1.19i·7^-s − 0.743·11^-s + 1.50i·13^-s + 1.04i·17^-s + 1.32·19^-s − 0.977i·23^-s + (−0.994 + 0.104i)25^-s + 0.439·29^-s − 0.794·31^-s + (1.19 − 0.0622i)35^-s + 0.267i·37^-s − 0.362·41^-s − 1.77i·43^-s + 1.36i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0521i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1440$ = $2^{5} \cdot 3^{2} \cdot 5$
Sign:	$-0.998 + 0.0521i$
Analytic conductor:	$84.9627$
Root analytic conductor:	$9.21752$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1440} (289, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1440,\ (\ :3/2),\ -0.998 + 0.0521i)$

Particular Values

$L(2)$	$\approx$	$0.4309510271$
$L(\frac12)$	$\approx$	$0.4309510271$
$L(\frac{5}{2})$		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 + (-0.582 - 11.1i)T $
good	7	$ 1 + 22.1iT - 343T^{2} $
	11	$ 1 + 27.1T + 1.33e3T^{2} $
	13	$ 1 - 70.3iT - 2.19e3T^{2} $
	17	$ 1 - 73.3iT - 4.91e3T^{2} $
	19	$ 1 - 110.T + 6.85e3T^{2} $
	23	$ 1 + 107. iT - 1.21e4T^{2} $
	29	$ 1 - 68.6T + 2.43e4T^{2} $
	31	$ 1 + 137.T + 2.97e4T^{2} $
	37	$ 1 - 60.3iT - 5.06e4T^{2} $
	41	$ 1 + 95.1T + 6.89e4T^{2} $
	43	$ 1 + 501. iT - 7.95e4T^{2} $
	47	$ 1 - 439. iT - 1.03e5T^{2} $
	53	$ 1 + 286. iT - 1.48e5T^{2} $
	59	$ 1 + 547.T + 2.05e5T^{2} $
	61	$ 1 - 511.T + 2.26e5T^{2} $
	67	$ 1 - 301. iT - 3.00e5T^{2} $
	71	$ 1 + 82.8T + 3.57e5T^{2} $
	73	$ 1 - 763. iT - 3.89e5T^{2} $
	79	$ 1 + 1.01e3T + 4.93e5T^{2} $
	83	$ 1 - 704. iT - 5.71e5T^{2} $
	89	$ 1 + 743.T + 7.04e5T^{2} $
	97	$ 1 + 1.13e3iT - 9.12e5T^{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

−9.820433557035771103141584654242, −8.743691665195022193356035803930, −7.76953964089059280314644539067, −7.07926857829208239512574675574, −6.56604357634855116773313171110, −5.49404130681564684187843836533, −4.31457648559671367350587126543, −3.65247142662642091508685185199, −2.55160986845160036651121839954, −1.41080830072191865187325676245, 0.098315548024506068386937757397, 1.26115625720937493011957772521, 2.58411660415548058652127806101, 3.35173137053809364079460227366, 4.91065232096615692656835329767, 5.35577404288576349444469087053, 5.91551533278749983652965557347, 7.43182573958730436632750498175, 7.951351698685359383766460806705, 8.809621410738761997033218630932

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.4309510271\)
\(L(\frac12)\)	\(\approx\)	\(0.4309510271\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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