LMFDB - L-function 2-12e3-24.11-c1-0-1

Dirichlet series

L(s) = 1

− 4.18·5^-s + 5.24i·7^-s + 3.16i·11^-s + 12.4·25^-s − 10.3·29^-s + 0.757i·31^-s − 21.9i·35^-s − 20.4·49^-s + 10.5·53^-s − 13.2i·55^-s − 10.3i·59^-s − 1.48·73^-s − 16.6·77^-s − 10i·79^-s − 13.5i·83^-s + ⋯

L(s) = 1

− 1.87·5^-s + 1.98i·7^-s + 0.954i·11^-s + 2.49·25^-s − 1.92·29^-s + 0.136i·31^-s − 3.70i·35^-s − 2.92·49^-s + 1.44·53^-s − 1.78i·55^-s − 1.35i·59^-s − 0.173·73^-s − 1.89·77^-s − 1.12i·79^-s − 1.48i·83^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$-0.707 + 0.707i$
Analytic conductor:	$13.7981$
Root analytic conductor:	$3.71458$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (863, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :1/2),\ -0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$0.2131522497$
$L(\frac12)$	$\approx$	$0.2131522497$
$L(\frac{3}{2})$		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 + 4.18T + 5T^{2} $
	7	$ 1 - 5.24iT - 7T^{2} $
	11	$ 1 - 3.16iT - 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 - 17T^{2} $
	19	$ 1 + 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 + 10.3T + 29T^{2} $
	31	$ 1 - 0.757iT - 31T^{2} $
	37	$ 1 - 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 + 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 10.5T + 53T^{2} $
	59	$ 1 + 10.3iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 + 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 1.48T + 73T^{2} $
	79	$ 1 + 10iT - 79T^{2} $
	83	$ 1 + 13.5iT - 83T^{2} $
	89	$ 1 - 89T^{2} $
	97	$ 1 + 17.9T + 97T^{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

−9.547656885111915666325380619690, −8.934635312467924722674841307436, −8.228464189015613996647022285983, −7.56983474544004762459998308515, −6.77927885653013895453276844571, −5.64785300870875321635008622012, −4.88938385489288922406673416075, −3.96035286650184237615237025471, −3.03881267763968231959156789020, −1.97670375471459973610302788714, 0.096981138953401852999747934456, 1.04979217871838222188891443650, 3.13298603663865883417467757297, 3.97580165522743888577883561158, 4.17452692712845990411526454924, 5.46084450413985307113292403857, 6.83139445651975501352426230041, 7.29515478831062056414062732431, 7.930518951082319733680449167457, 8.549712214629354446632092106822

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.2131522497\)
\(L(\frac12)\)	\(\approx\)	\(0.2131522497\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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