LMFDB - L-function 2-60e2-3.2-c2-0-4

Dirichlet series

L(s) = 1

− 4·7^-s − 16.9i·11^-s − 8·13^-s + 12.7i·17^-s + 16·19^-s − 16.9i·23^-s + 4.24i·29^-s − 44·31^-s + 34·37^-s + 46.6i·41^-s − 40·43^-s − 84.8i·47^-s − 33·49^-s − 38.1i·53^-s − 33.9i·59^-s + ⋯

L(s) = 1

− 0.571·7^-s − 1.54i·11^-s − 0.615·13^-s + 0.748i·17^-s + 0.842·19^-s − 0.737i·23^-s + 0.146i·29^-s − 1.41·31^-s + 0.918·37^-s + 1.13i·41^-s − 0.930·43^-s − 1.80i·47^-s − 0.673·49^-s − 0.720i·53^-s − 0.575i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3600$ = $2^{4} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.577 - 0.816i$
Analytic conductor:	$98.0928$
Root analytic conductor:	$9.90418$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3600} (1601, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3600,\ (\ :1),\ -0.577 - 0.816i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.3524258511$
$L(\frac12)$	$\approx$	$0.3524258511$
$L(2)$		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 $
good	7	$ 1 + 4T + 49T^{2} $
	11	$ 1 + 16.9iT - 121T^{2} $
	13	$ 1 + 8T + 169T^{2} $
	17	$ 1 - 12.7iT - 289T^{2} $
	19	$ 1 - 16T + 361T^{2} $
	23	$ 1 + 16.9iT - 529T^{2} $
	29	$ 1 - 4.24iT - 841T^{2} $
	31	$ 1 + 44T + 961T^{2} $
	37	$ 1 - 34T + 1.36e3T^{2} $
	41	$ 1 - 46.6iT - 1.68e3T^{2} $
	43	$ 1 + 40T + 1.84e3T^{2} $
	47	$ 1 + 84.8iT - 2.20e3T^{2} $
	53	$ 1 + 38.1iT - 2.80e3T^{2} $
	59	$ 1 + 33.9iT - 3.48e3T^{2} $
	61	$ 1 - 50T + 3.72e3T^{2} $
	67	$ 1 - 8T + 4.48e3T^{2} $
	71	$ 1 - 50.9iT - 5.04e3T^{2} $
	73	$ 1 - 16T + 5.32e3T^{2} $
	79	$ 1 - 76T + 6.24e3T^{2} $
	83	$ 1 - 118. iT - 6.88e3T^{2} $
	89	$ 1 - 12.7iT - 7.92e3T^{2} $
	97	$ 1 + 176T + 9.40e3T^{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

−8.470684268293794445005013037152, −8.107354448560472577558314763376, −7.04612330417815565164907585653, −6.42616408731789284046910228023, −5.65228552912131393858021600377, −4.99241934142009144001226198015, −3.76521328297017904724221582558, −3.26986400289740248268808580733, −2.25583513021034354290234889279, −0.953922331406027318708089208768, 0.085182925324381150262690812678, 1.51137411889602613917537822818, 2.48304934420253975741794279409, 3.35451403436282352298456701524, 4.34470150617042283557504613098, 5.06331176541870079892236774026, 5.80964838702521666748837225089, 6.86217240192479193474016579466, 7.33770386476299599369825208911, 7.87910268859627431230676837034

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.3524258511\)
\(L(\frac12)\)	\(\approx\)	\(0.3524258511\)
\(L(2)\)		not available
\(L(1)\)		not available

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