LMFDB - L-function 2-6048-12.11-c1-0-8

Dirichlet series

L(s) = 1

− 0.317i·5^-s + i·7^-s + 1.41·11^-s − 4.44·13^-s − 5.97i·17^-s + 2.44i·19^-s − 5.65·23^-s + 4.89·25^-s + 5.51i·29^-s − 6.44i·31^-s + 0.317·35^-s + 9·37^-s + 7.24i·41^-s + 6.34i·43^-s − 10.8·47^-s + ⋯

L(s) = 1

− 0.142i·5^-s + 0.377i·7^-s + 0.426·11^-s − 1.23·13^-s − 1.44i·17^-s + 0.561i·19^-s − 1.17·23^-s + 0.979·25^-s + 1.02i·29^-s − 1.15i·31^-s + 0.0537·35^-s + 1.47·37^-s + 1.13i·41^-s + 0.968i·43^-s − 1.58·47^-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

$L(1)$	$\approx$	$0.6274522668$
$L(\frac12)$	$\approx$	$0.6274522668$
$L(\frac{3}{2})$		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 + 0.317iT - 5T^{2} $
	11	$ 1 - 1.41T + 11T^{2} $
	13	$ 1 + 4.44T + 13T^{2} $
	17	$ 1 + 5.97iT - 17T^{2} $
	19	$ 1 - 2.44iT - 19T^{2} $
	23	$ 1 + 5.65T + 23T^{2} $
	29	$ 1 - 5.51iT - 29T^{2} $
	31	$ 1 + 6.44iT - 31T^{2} $
	37	$ 1 - 9T + 37T^{2} $
	41	$ 1 - 7.24iT - 41T^{2} $
	43	$ 1 - 6.34iT - 43T^{2} $
	47	$ 1 + 10.8T + 47T^{2} $
	53	$ 1 + 11.9iT - 53T^{2} $
	59	$ 1 - 5.83T + 59T^{2} $
	61	$ 1 + 10.4T + 61T^{2} $
	67	$ 1 - 5.79iT - 67T^{2} $
	71	$ 1 - 5.65T + 71T^{2} $
	73	$ 1 + 9.79T + 73T^{2} $
	79	$ 1 - 14.3iT - 79T^{2} $
	83	$ 1 - 8.02T + 83T^{2} $
	89	$ 1 - 15.4iT - 89T^{2} $
	97	$ 1 + 14.2T + 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

−8.134129382718348448390291473053, −7.79168154611401389226191866399, −6.84567328571023144385413617798, −6.33107897416134227291517370974, −5.32553435020339519115285291831, −4.84470315280512606941124817371, −4.02372797656701503200510039004, −2.95279987940488461058959726773, −2.34035872810408318314180758878, −1.16258443914382381030516145491, 0.16263994893559228941018338420, 1.51267564013525909546234150124, 2.41809082446386980084621520154, 3.34584182821393329618124526860, 4.24263099620280640355717897319, 4.74937857605992617712100664139, 5.79698560399343750598438187507, 6.39171373799039633449429844258, 7.14536998319702744695221369150, 7.74800446019954675979808087438

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6274522668\)
\(L(\frac12)\)	\(\approx\)	\(0.6274522668\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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