LMFDB - L-function 2-1040-13.12-c3-0-74

Dirichlet series

L(s) = 1

− 8.59·3^-s − 5i·5^-s − 19.4i·7^-s + 46.8·9^-s − 20.0i·11^-s + (−46.8 − 0.655i)13^-s + 42.9i·15^-s − 89.4·17^-s − 11.7i·19^-s + 167. i·21^-s − 157.·23^-s − 25·25^-s − 170.·27^-s + 116.·29^-s − 245. i·31^-s + ⋯

L(s) = 1

− 1.65·3^-s − 0.447i·5^-s − 1.05i·7^-s + 1.73·9^-s − 0.550i·11^-s + (−0.999 − 0.0139i)13^-s + 0.739i·15^-s − 1.27·17^-s − 0.142i·19^-s + 1.73i·21^-s − 1.42·23^-s − 0.200·25^-s − 1.21·27^-s + 0.743·29^-s − 1.42i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0139 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0139 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1040$ = $2^{4} \cdot 5 \cdot 13$
Sign:	$0.0139 - 0.999i$
Analytic conductor:	$61.3619$
Root analytic conductor:	$7.83338$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1040} (961, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1040,\ (\ :3/2),\ 0.0139 - 0.999i)$

Particular Values

$L(2)$	$\approx$	$0.1659620220$
$L(\frac12)$	$\approx$	$0.1659620220$
$L(\frac{5}{2})$		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 $
	5	$ 1 + 5iT $
	13	$ 1 + (46.8 + 0.655i)T $
good	3	$ 1 + 8.59T + 27T^{2} $
	7	$ 1 + 19.4iT - 343T^{2} $
	11	$ 1 + 20.0iT - 1.33e3T^{2} $
	17	$ 1 + 89.4T + 4.91e3T^{2} $
	19	$ 1 + 11.7iT - 6.85e3T^{2} $
	23	$ 1 + 157.T + 1.21e4T^{2} $
	29	$ 1 - 116.T + 2.43e4T^{2} $
	31	$ 1 + 245. iT - 2.97e4T^{2} $
	37	$ 1 + 213. iT - 5.06e4T^{2} $
	41	$ 1 + 442. iT - 6.89e4T^{2} $
	43	$ 1 + 184.T + 7.95e4T^{2} $
	47	$ 1 + 247. iT - 1.03e5T^{2} $
	53	$ 1 - 14.9T + 1.48e5T^{2} $
	59	$ 1 + 416. iT - 2.05e5T^{2} $
	61	$ 1 + 898.T + 2.26e5T^{2} $
	67	$ 1 - 847. iT - 3.00e5T^{2} $
	71	$ 1 + 512. iT - 3.57e5T^{2} $
	73	$ 1 + 533. iT - 3.89e5T^{2} $
	79	$ 1 - 90.0T + 4.93e5T^{2} $
	83	$ 1 - 1.09e3iT - 5.71e5T^{2} $
	89	$ 1 - 117. iT - 7.04e5T^{2} $
	97	$ 1 + 895. iT - 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

−9.068472839338840378150324927980, −7.86551034289197242061288070150, −7.06105545730740342887152377366, −6.29200691781595268151200372512, −5.48300216323861555245113842855, −4.58089387579395447126152875606, −3.96935153156182183910337896064, −2.06062971849731524508591292163, −0.58248411240011044656906205722, −0.090975624011096562780655026149, 1.65697525183573201211801221260, 2.79037025209922008540562116363, 4.50344006590354113436896545224, 4.97676277619462261263668994987, 6.07793029603192012593553586867, 6.49632600359812025565907181847, 7.40558096198995508422007611526, 8.523739458842951169650019948504, 9.646433851378708073036296300405, 10.24820462028198914504433303677

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

\(L(2)\)	\(\approx\)	\(0.1659620220\)
\(L(\frac12)\)	\(\approx\)	\(0.1659620220\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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