LMFDB - L-function 2-6048-12.11-c1-0-29

Dirichlet series

L(s) = 1

− 2.96i·5^-s + i·7^-s + 4.76·11^-s − 1.26·13^-s + 0.378i·17^-s + 5i·19^-s − 1.93·23^-s − 3.80·25^-s − 4.24i·29^-s + 10.4i·31^-s + 2.96·35^-s + 2.46·37^-s + 10.6i·41^-s + 4.73i·43^-s − 3.20·47^-s + ⋯

L(s) = 1

− 1.32i·5^-s + 0.377i·7^-s + 1.43·11^-s − 0.351·13^-s + 0.0919i·17^-s + 1.14i·19^-s − 0.402·23^-s − 0.760·25^-s − 0.787i·29^-s + 1.87i·31^-s + 0.501·35^-s + 0.405·37^-s + 1.67i·41^-s + 0.721i·43^-s − 0.467·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$48.2935$
Root analytic conductor:	$6.94935$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6048} (2591, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.747551332$
$L(\frac12)$	$\approx$	$1.747551332$
$L(\frac{3}{2})$		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 $
	7	$ 1 - iT $
good	5	$ 1 + 2.96iT - 5T^{2} $
	11	$ 1 - 4.76T + 11T^{2} $
	13	$ 1 + 1.26T + 13T^{2} $
	17	$ 1 - 0.378iT - 17T^{2} $
	19	$ 1 - 5iT - 19T^{2} $
	23	$ 1 + 1.93T + 23T^{2} $
	29	$ 1 + 4.24iT - 29T^{2} $
	31	$ 1 - 10.4iT - 31T^{2} $
	37	$ 1 - 2.46T + 37T^{2} $
	41	$ 1 - 10.6iT - 41T^{2} $
	43	$ 1 - 4.73iT - 43T^{2} $
	47	$ 1 + 3.20T + 47T^{2} $
	53	$ 1 - 8.76iT - 53T^{2} $
	59	$ 1 - 3.48T + 59T^{2} $
	61	$ 1 + 4.92T + 61T^{2} $
	67	$ 1 - 7.66iT - 67T^{2} $
	71	$ 1 + 0.240T + 71T^{2} $
	73	$ 1 + 0.196T + 73T^{2} $
	79	$ 1 - 1.80iT - 79T^{2} $
	83	$ 1 - 7.82T + 83T^{2} $
	89	$ 1 + 1.93iT - 89T^{2} $
	97	$ 1 + 12.9T + 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

−8.316702068094678836154287622550, −7.61921361094488690274402441811, −6.57816808628782255827449789560, −6.05425354985838043138026660756, −5.24175679425701473186818272836, −4.51554240016273079845437399216, −3.94349971315046700356563912683, −2.92862919404095821200058885417, −1.64136816663485923068009508528, −1.12688807482298077541878111338, 0.47875312469303823251492868929, 1.87367349928948154386784726585, 2.68169178109197688005205847913, 3.60336209200955570904301723961, 4.11293215343910767831589082793, 5.11143195950984289809448776190, 6.09069365423354511818776610071, 6.70246270582561584343158834358, 7.11930006615749208937647147579, 7.77366506703343517336105968348

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.747551332\)
\(L(\frac12)\)	\(\approx\)	\(1.747551332\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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