LMFDB - L-function 2-12e3-24.11-c3-0-72

Dirichlet series

L(s) = 1

+ 12.8·5^-s + 17.2i·7^-s − 28.7i·11^-s − 33.2i·13^-s + 5.37i·17^-s − 50.9·19^-s + 210.·23^-s + 39.4·25^-s − 83.1·29^-s + 40.4i·31^-s + 220. i·35^-s − 264. i·37^-s − 399. i·41^-s − 208.·43^-s + 178.·47^-s + ⋯

L(s) = 1

+ 1.14·5^-s + 0.929i·7^-s − 0.787i·11^-s − 0.709i·13^-s + 0.0766i·17^-s − 0.615·19^-s + 1.90·23^-s + 0.315·25^-s − 0.532·29^-s + 0.234i·31^-s + 1.06i·35^-s − 1.17i·37^-s − 1.52i·41^-s − 0.737·43^-s + 0.553·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$101.955$
Root analytic conductor:	$10.0972$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (863, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :3/2),\ 0.707 + 0.707i)$

Particular Values

$L(2)$	$\approx$	$2.613624670$
$L(\frac12)$	$\approx$	$2.613624670$
$L(\frac{5}{2})$		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 $
good	5	$ 1 - 12.8T + 125T^{2} $
	7	$ 1 - 17.2iT - 343T^{2} $
	11	$ 1 + 28.7iT - 1.33e3T^{2} $
	13	$ 1 + 33.2iT - 2.19e3T^{2} $
	17	$ 1 - 5.37iT - 4.91e3T^{2} $
	19	$ 1 + 50.9T + 6.85e3T^{2} $
	23	$ 1 - 210.T + 1.21e4T^{2} $
	29	$ 1 + 83.1T + 2.43e4T^{2} $
	31	$ 1 - 40.4iT - 2.97e4T^{2} $
	37	$ 1 + 264. iT - 5.06e4T^{2} $
	41	$ 1 + 399. iT - 6.89e4T^{2} $
	43	$ 1 + 208.T + 7.95e4T^{2} $
	47	$ 1 - 178.T + 1.03e5T^{2} $
	53	$ 1 - 82.3T + 1.48e5T^{2} $
	59	$ 1 + 554. iT - 2.05e5T^{2} $
	61	$ 1 + 349. iT - 2.26e5T^{2} $
	67	$ 1 - 664.T + 3.00e5T^{2} $
	71	$ 1 - 399.T + 3.57e5T^{2} $
	73	$ 1 + 802.T + 3.89e5T^{2} $
	79	$ 1 - 322. iT - 4.93e5T^{2} $
	83	$ 1 - 148. iT - 5.71e5T^{2} $
	89	$ 1 + 826. iT - 7.04e5T^{2} $
	97	$ 1 - 1.11e3T + 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

−8.967997773280304145400887846172, −8.296911254293129918101960729198, −7.17194048141650759224905050454, −6.28610491081094941347305870800, −5.54796722721173476968830284582, −5.14831561243561913631782934018, −3.64881293630127488290146376532, −2.68399754911891455890774556358, −1.89533617226338844741548794462, −0.59156762371269290862235023669, 1.05514191658952967580723594423, 1.91891384588047210690602733452, 2.97130191888838003336740596965, 4.22821855218847328179029953015, 4.88323799050056210032556027365, 5.88386789206884284179069779957, 6.81109108151141526657591166158, 7.19220007157985245149233640633, 8.348130478964730574943029233253, 9.256869441938675370005524406648

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.613624670\)
\(L(\frac12)\)	\(\approx\)	\(2.613624670\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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