LMFDB - L-function 2-720-15.14-c2-0-17

Dirichlet series

L(s) = 1

+ (4.79 − 1.41i)5^-s + 6.78i·7^-s − 9.89i·11^-s − 20.3i·13^-s + 19.1·17^-s − 12·19^-s − 9.59·23^-s + (20.9 − 13.5i)25^-s − 8.48i·29^-s + 38·31^-s + (9.59 + 32.5i)35^-s − 6.78i·37^-s + 69.2i·41^-s − 67.8i·43^-s + 76.7·47^-s + ⋯

L(s) = 1

+ (0.959 − 0.282i)5^-s + 0.968i·7^-s − 0.899i·11^-s − 1.56i·13^-s + 1.12·17^-s − 0.631·19^-s − 0.417·23^-s + (0.839 − 0.542i)25^-s − 0.292i·29^-s + 1.22·31^-s + (0.274 + 0.929i)35^-s − 0.183i·37^-s + 1.69i·41^-s − 1.57i·43^-s + 1.63·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$720$ = $2^{4} \cdot 3^{2} \cdot 5$
Sign:	$0.784 + 0.619i$
Analytic conductor:	$19.6185$
Root analytic conductor:	$4.42928$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{720} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 720,\ (\ :1),\ 0.784 + 0.619i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.192943079$
$L(\frac12)$	$\approx$	$2.192943079$
$L(2)$		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 + (-4.79 + 1.41i)T $
good	7	$ 1 - 6.78iT - 49T^{2} $
	11	$ 1 + 9.89iT - 121T^{2} $
	13	$ 1 + 20.3iT - 169T^{2} $
	17	$ 1 - 19.1T + 289T^{2} $
	19	$ 1 + 12T + 361T^{2} $
	23	$ 1 + 9.59T + 529T^{2} $
	29	$ 1 + 8.48iT - 841T^{2} $
	31	$ 1 - 38T + 961T^{2} $
	37	$ 1 + 6.78iT - 1.36e3T^{2} $
	41	$ 1 - 69.2iT - 1.68e3T^{2} $
	43	$ 1 + 67.8iT - 1.84e3T^{2} $
	47	$ 1 - 76.7T + 2.20e3T^{2} $
	53	$ 1 + 2.80e3T^{2} $
	59	$ 1 - 83.4iT - 3.48e3T^{2} $
	61	$ 1 + 70T + 3.72e3T^{2} $
	67	$ 1 + 108. iT - 4.48e3T^{2} $
	71	$ 1 + 118. iT - 5.04e3T^{2} $
	73	$ 1 + 13.5iT - 5.32e3T^{2} $
	79	$ 1 + 30T + 6.24e3T^{2} $
	83	$ 1 - 134.T + 6.88e3T^{2} $
	89	$ 1 - 32.5iT - 7.92e3T^{2} $
	97	$ 1 + 94.9iT - 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

−10.19460090177725516875601497371, −9.211392793951706771916775424959, −8.468978167333125006970051298344, −7.73247211469541238096452701619, −6.09555595944207973957427994680, −5.85056945671979228775462136859, −4.90483348621287154714825787387, −3.28384711569160241651586701261, −2.38219398738874057388919931427, −0.862865440054364603715582903708, 1.32603783672534691073634855723, 2.40863458462147401742208820245, 3.90377745000679921489198038403, 4.74751397708240029935046010874, 5.97881284645072039736016796429, 6.83708201655075239321110343827, 7.46010812417736448298208718025, 8.715782582436815930760142683800, 9.683561682656825836181478632967, 10.14096636408174166776857100530

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.192943079\)
\(L(\frac12)\)	\(\approx\)	\(2.192943079\)
\(L(2)\)		not available
\(L(1)\)		not available

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