LMFDB - L-function 2-6048-12.11-c1-0-43

Dirichlet series

L(s) = 1

− 3.24i·5^-s + i·7^-s − 6.07·11^-s + 0.0963·13^-s + 3.41i·17^-s + 5.89i·19^-s + 7.54·23^-s − 5.56·25^-s − 5.81i·29^-s − 1.07i·31^-s + 3.24·35^-s + 7.82·37^-s + 0.614i·41^-s + 7.75i·43^-s + 0.585·47^-s + ⋯

L(s) = 1

− 1.45i·5^-s + 0.377i·7^-s − 1.83·11^-s + 0.0267·13^-s + 0.828i·17^-s + 1.35i·19^-s + 1.57·23^-s − 1.11·25^-s − 1.07i·29^-s − 0.192i·31^-s + 0.549·35^-s + 1.28·37^-s + 0.0958i·41^-s + 1.18i·43^-s + 0.0854·47^-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

$L(1)$	$\approx$	$1.548022118$
$L(\frac12)$	$\approx$	$1.548022118$
$L(\frac{3}{2})$		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 + 3.24iT - 5T^{2} $
	11	$ 1 + 6.07T + 11T^{2} $
	13	$ 1 - 0.0963T + 13T^{2} $
	17	$ 1 - 3.41iT - 17T^{2} $
	19	$ 1 - 5.89iT - 19T^{2} $
	23	$ 1 - 7.54T + 23T^{2} $
	29	$ 1 + 5.81iT - 29T^{2} $
	31	$ 1 + 1.07iT - 31T^{2} $
	37	$ 1 - 7.82T + 37T^{2} $
	41	$ 1 - 0.614iT - 41T^{2} $
	43	$ 1 - 7.75iT - 43T^{2} $
	47	$ 1 - 0.585T + 47T^{2} $
	53	$ 1 + 7.39iT - 53T^{2} $
	59	$ 1 + 9.01T + 59T^{2} $
	61	$ 1 + 8.92T + 61T^{2} $
	67	$ 1 + 10.4iT - 67T^{2} $
	71	$ 1 - 9.19T + 71T^{2} $
	73	$ 1 - 11.8T + 73T^{2} $
	79	$ 1 - 11.0iT - 79T^{2} $
	83	$ 1 - 15.6T + 83T^{2} $
	89	$ 1 + 3.01iT - 89T^{2} $
	97	$ 1 - 18.5T + 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

−7.981312263550060297030100655935, −7.71121968376370051235180430674, −6.32881846679603098267295586294, −5.75985837076499415046225789550, −5.02388510262064869326466260908, −4.62170401685743578915797683343, −3.56568024367050850623630559081, −2.58790426217531541825395658355, −1.66746313885331905129269781127, −0.59000010012690945533322683630, 0.70143804855573540336420300465, 2.37292424376077574886755864900, 2.84064677557649676571379900505, 3.42664841136651878656592882771, 4.76872023318225199712368887438, 5.13841283104284438584973739379, 6.16645039240651936337803750793, 6.97403847626819071813965293664, 7.32239841020660246860016097628, 7.87741362493891456817013076962

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.548022118\)
\(L(\frac12)\)	\(\approx\)	\(1.548022118\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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