LMFDB - L-function 2-12e3-24.5-c2-0-24

Dirichlet series

L(s) = 1

− 8.66·7^-s + 12.1i·13^-s − 37i·19^-s − 25·25^-s + 41.5·31^-s + 12.1i·37^-s + 22i·43^-s + 26.0·49^-s + 112. i·61^-s + 13i·67^-s + 143·73^-s + 157.·79^-s − 105. i·91^-s + 169·97^-s + 133.·103^-s + ⋯

L(s) = 1

− 1.23·7^-s + 0.932i·13^-s − 1.94i·19^-s − 25^-s + 1.34·31^-s + 0.327i·37^-s + 0.511i·43^-s + 0.530·49^-s + 1.84i·61^-s + 0.194i·67^-s + 1.95·73^-s + 1.99·79^-s − 1.15i·91^-s + 1.74·97^-s + 1.29·103^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$0.965 - 0.258i$
Analytic conductor:	$47.0845$
Root analytic conductor:	$6.86182$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (161, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :1),\ 0.965 - 0.258i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.387839428$
$L(\frac12)$	$\approx$	$1.387839428$
$L(2)$		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 $
good	5	$ 1 + 25T^{2} $
	7	$ 1 + 8.66T + 49T^{2} $
	11	$ 1 + 121T^{2} $
	13	$ 1 - 12.1iT - 169T^{2} $
	17	$ 1 - 289T^{2} $
	19	$ 1 + 37iT - 361T^{2} $
	23	$ 1 - 529T^{2} $
	29	$ 1 + 841T^{2} $
	31	$ 1 - 41.5T + 961T^{2} $
	37	$ 1 - 12.1iT - 1.36e3T^{2} $
	41	$ 1 - 1.68e3T^{2} $
	43	$ 1 - 22iT - 1.84e3T^{2} $
	47	$ 1 - 2.20e3T^{2} $
	53	$ 1 + 2.80e3T^{2} $
	59	$ 1 + 3.48e3T^{2} $
	61	$ 1 - 112. iT - 3.72e3T^{2} $
	67	$ 1 - 13iT - 4.48e3T^{2} $
	71	$ 1 - 5.04e3T^{2} $
	73	$ 1 - 143T + 5.32e3T^{2} $
	79	$ 1 - 157.T + 6.24e3T^{2} $
	83	$ 1 + 6.88e3T^{2} $
	89	$ 1 - 7.92e3T^{2} $
	97	$ 1 - 169T + 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

−9.316338419534469682506743877935, −8.521006900322470495891544280141, −7.44990377553789781876627592530, −6.65524473491085927544021907969, −6.20858863901102755526189719398, −4.99880050061949363281868860836, −4.16175749515988567105342992924, −3.13468168023730796251444903146, −2.26786514126778240985642605844, −0.68365569829573108571757791970, 0.58175494189135854683974143364, 2.07287530495260811699206850914, 3.28320498975150845609546803571, 3.81509397312334085542408966311, 5.12359295567212659393624673233, 6.03927362472384893100281590632, 6.48645914352682859191780861891, 7.69826721122053408278412310135, 8.151271335384851895718940192430, 9.241921150800553605111619486064

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.387839428\)
\(L(\frac12)\)	\(\approx\)	\(1.387839428\)
\(L(2)\)		not available
\(L(1)\)		not available

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