LMFDB - L-function 2-1512-21.20-c1-0-13

Dirichlet series

L(s) = 1

− 4.29·5^-s + (0.832 − 2.51i)7^-s + 2.51i·11^-s + 5.36i·13^-s − 3.29·17^-s − 0.663i·19^-s − 6.44i·23^-s + 13.4·25^-s + 4.26i·29^-s − 4.78i·31^-s + (−3.58 + 10.7i)35^-s − 0.746·37^-s + 7.99·41^-s + 7.69·43^-s + 9.32·47^-s + ⋯

L(s) = 1

− 1.92·5^-s + (0.314 − 0.949i)7^-s + 0.758i·11^-s + 1.48i·13^-s − 0.798·17^-s − 0.152i·19^-s − 1.34i·23^-s + 2.69·25^-s + 0.791i·29^-s − 0.860i·31^-s + (−0.605 + 1.82i)35^-s − 0.122·37^-s + 1.24·41^-s + 1.17·43^-s + 1.35·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1512$ = $2^{3} \cdot 3^{3} \cdot 7$
Sign:	$0.949 + 0.314i$
Analytic conductor:	$12.0733$
Root analytic conductor:	$3.47467$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1512} (377, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1512,\ (\ :1/2),\ 0.949 + 0.314i)$

Particular Values

$L(1)$	$\approx$	$1.004047652$
$L(\frac12)$	$\approx$	$1.004047652$
$L(\frac{3}{2})$		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 $
	7	$ 1 + (-0.832 + 2.51i)T $
good	5	$ 1 + 4.29T + 5T^{2} $
	11	$ 1 - 2.51iT - 11T^{2} $
	13	$ 1 - 5.36iT - 13T^{2} $
	17	$ 1 + 3.29T + 17T^{2} $
	19	$ 1 + 0.663iT - 19T^{2} $
	23	$ 1 + 6.44iT - 23T^{2} $
	29	$ 1 - 4.26iT - 29T^{2} $
	31	$ 1 + 4.78iT - 31T^{2} $
	37	$ 1 + 0.746T + 37T^{2} $
	41	$ 1 - 7.99T + 41T^{2} $
	43	$ 1 - 7.69T + 43T^{2} $
	47	$ 1 - 9.32T + 47T^{2} $
	53	$ 1 + 11.8iT - 53T^{2} $
	59	$ 1 - 1.15T + 59T^{2} $
	61	$ 1 + 10.7iT - 61T^{2} $
	67	$ 1 - 0.587T + 67T^{2} $
	71	$ 1 - 9.77iT - 71T^{2} $
	73	$ 1 - 12.6iT - 73T^{2} $
	79	$ 1 - 11.1T + 79T^{2} $
	83	$ 1 - 7.07T + 83T^{2} $
	89	$ 1 - 3.25T + 89T^{2} $
	97	$ 1 + 10.8iT - 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

−9.263054816520121646362966927183, −8.545124971740363486952226489141, −7.72615431611046323640634057266, −7.08881204252985359527906098912, −6.60854641003400280794135395591, −4.80360846669442080795766480136, −4.27748146861632371516865371797, −3.79710701924596235780296857879, −2.29542689015355535862452036218, −0.64793024904421314389048694868, 0.75989560562948769710410682951, 2.69068078189641922203147676246, 3.47515171954064768090935137418, 4.37551570243018983080593834783, 5.36721250699113445421259208702, 6.15692833069205539811010785540, 7.60416044889611298720749281416, 7.71555709274488779412057196594, 8.665861550557762400620561087174, 9.170175657454652202025811251420

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(1.004047652\)
\(L(\frac12)\)	\(\approx\)	\(1.004047652\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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