LMFDB - L-function 2-6048-8.5-c1-0-88

Dirichlet series

L(s) = 1

− 3.06i·5^-s + 7^-s − 5.80i·11^-s − 3.52i·13^-s + 6.79·17^-s − 5.28i·19^-s − 5.65·23^-s − 4.41·25^-s − 1.21i·29^-s − 0.107·31^-s − 3.06i·35^-s − 4.90i·37^-s + 11.6·41^-s + 1.85i·43^-s + 6.76·47^-s + ⋯

L(s) = 1

− 1.37i·5^-s + 0.377·7^-s − 1.75i·11^-s − 0.977i·13^-s + 1.64·17^-s − 1.21i·19^-s − 1.17·23^-s − 0.883·25^-s − 0.225i·29^-s − 0.0192·31^-s − 0.518i·35^-s − 0.805i·37^-s + 1.82·41^-s + 0.283i·43^-s + 0.986·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 + 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$-0.858 + 0.513i$
Analytic conductor:	$48.2935$
Root analytic conductor:	$6.94935$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6048} (3025, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6048,\ (\ :1/2),\ -0.858 + 0.513i)$

Particular Values

$L(1)$	$\approx$	$2.099727064$
$L(\frac12)$	$\approx$	$2.099727064$
$L(\frac{3}{2})$		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 - T $
good	5	$ 1 + 3.06iT - 5T^{2} $
	11	$ 1 + 5.80iT - 11T^{2} $
	13	$ 1 + 3.52iT - 13T^{2} $
	17	$ 1 - 6.79T + 17T^{2} $
	19	$ 1 + 5.28iT - 19T^{2} $
	23	$ 1 + 5.65T + 23T^{2} $
	29	$ 1 + 1.21iT - 29T^{2} $
	31	$ 1 + 0.107T + 31T^{2} $
	37	$ 1 + 4.90iT - 37T^{2} $
	41	$ 1 - 11.6T + 41T^{2} $
	43	$ 1 - 1.85iT - 43T^{2} $
	47	$ 1 - 6.76T + 47T^{2} $
	53	$ 1 + 11.4iT - 53T^{2} $
	59	$ 1 + 7.85iT - 59T^{2} $
	61	$ 1 - 12.0iT - 61T^{2} $
	67	$ 1 - 6.66iT - 67T^{2} $
	71	$ 1 - 1.98T + 71T^{2} $
	73	$ 1 + 6.52T + 73T^{2} $
	79	$ 1 + 2.30T + 79T^{2} $
	83	$ 1 - 12.5iT - 83T^{2} $
	89	$ 1 - 7.79T + 89T^{2} $
	97	$ 1 - 4.05T + 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

−8.063045673395509952652582751129, −7.29259110650929122645287269158, −5.95226256236727521622885837450, −5.67781624708471757336309236511, −5.05107523960616669557144887919, −4.12156319598232520449461399118, −3.37207156467672864500140103930, −2.42524928775423972724596232778, −1.00119368716194071473003383782, −0.64371242796815150789677727192, 1.52058632398388125349210657854, 2.18537927280537180198339739725, 3.12264427086933459314815240446, 4.01023008883300416014376981797, 4.57848900766140629009810680896, 5.72322504064145012681557026302, 6.22452720711417038889495616601, 7.16902824639950210094978951725, 7.49618454293350801034562874767, 8.068256659833425693985898802882

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.099727064\)
\(L(\frac12)\)	\(\approx\)	\(2.099727064\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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