LMFDB - L-function 2-1440-5.4-c3-0-36

Dirichlet series

L(s) = 1

+ (−8.58 − 7.16i)5^-s + 28.3i·7^-s + 65.2·11^-s + 33.6i·13^-s − 73.3i·17^-s + 134.·19^-s − 14.7i·23^-s + (22.3 + 122. i)25^-s − 224.·29^-s − 68.8·31^-s + (202. − 243. i)35^-s − 196. i·37^-s + 143.·41^-s − 15.0i·43^-s − 134. i·47^-s + ⋯

L(s) = 1

+ (−0.767 − 0.640i)5^-s + 1.52i·7^-s + 1.78·11^-s + 0.718i·13^-s − 1.04i·17^-s + 1.61·19^-s − 0.133i·23^-s + (0.178 + 0.983i)25^-s − 1.43·29^-s − 0.398·31^-s + (0.979 − 1.17i)35^-s − 0.872i·37^-s + 0.545·41^-s − 0.0534i·43^-s − 0.417i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1440$ = $2^{5} \cdot 3^{2} \cdot 5$
Sign:	$0.640 - 0.767i$
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:	$(2,\ 1440,\ (\ :3/2),\ 0.640 - 0.767i)$

Particular Values

$L(2)$	$\approx$	$1.998296291$
$L(\frac12)$	$\approx$	$1.998296291$
$L(\frac{5}{2})$		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 + (8.58 + 7.16i)T $
good	7	$ 1 - 28.3iT - 343T^{2} $
	11	$ 1 - 65.2T + 1.33e3T^{2} $
	13	$ 1 - 33.6iT - 2.19e3T^{2} $
	17	$ 1 + 73.3iT - 4.91e3T^{2} $
	19	$ 1 - 134.T + 6.85e3T^{2} $
	23	$ 1 + 14.7iT - 1.21e4T^{2} $
	29	$ 1 + 224.T + 2.43e4T^{2} $
	31	$ 1 + 68.8T + 2.97e4T^{2} $
	37	$ 1 + 196. iT - 5.06e4T^{2} $
	41	$ 1 - 143.T + 6.89e4T^{2} $
	43	$ 1 + 15.0iT - 7.95e4T^{2} $
	47	$ 1 + 134. iT - 1.03e5T^{2} $
	53	$ 1 - 262. iT - 1.48e5T^{2} $
	59	$ 1 - 119.T + 2.05e5T^{2} $
	61	$ 1 - 16.5T + 2.26e5T^{2} $
	67	$ 1 - 545. iT - 3.00e5T^{2} $
	71	$ 1 + 199.T + 3.57e5T^{2} $
	73	$ 1 + 43.2iT - 3.89e5T^{2} $
	79	$ 1 - 438.T + 4.93e5T^{2} $
	83	$ 1 - 1.22e3iT - 5.71e5T^{2} $
	89	$ 1 - 723.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

−9.189187629661858420910846663069, −8.795507949261063672295523240467, −7.62479217509087790643784072354, −6.94832739312943683883370088778, −5.83859592585846246485813520324, −5.17369215844217033729553371373, −4.14798981091642938559465305181, −3.31075046706515923633951982129, −2.02102699726267411937159830675, −0.916420813942254349472207768368, 0.60056120893212819174337816022, 1.53040152235286694191464169871, 3.42371333599497566894840840404, 3.65984965634900910398055854886, 4.58132878198676697965991392999, 5.93459929366125543139677946006, 6.78622486322649681410690078639, 7.43732265132176471546070038974, 7.974532122857053507935299634760, 9.128831560725129775758468624750

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

\(L(2)\)	\(\approx\)	\(1.998296291\)
\(L(\frac12)\)	\(\approx\)	\(1.998296291\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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