LMFDB - L-function 2-1344-56.27-c3-0-76

Dirichlet series

L(s) = 1

+ 3i·3^-s + 16.7·5^-s + (16.4 − 8.44i)7^-s − 9·9^-s − 8.98·11^-s + 6.28·13^-s + 50.1i·15^-s − 106. i·17^-s − 25.1i·19^-s + (25.3 + 49.4i)21^-s + 19.5i·23^-s + 154.·25^-s − 27i·27^-s + 132. i·29^-s + 120.·31^-s + ⋯

L(s) = 1

+ 0.577i·3^-s + 1.49·5^-s + (0.889 − 0.456i)7^-s − 0.333·9^-s − 0.246·11^-s + 0.134·13^-s + 0.862i·15^-s − 1.52i·17^-s − 0.304i·19^-s + (0.263 + 0.513i)21^-s + 0.177i·23^-s + 1.23·25^-s − 0.192i·27^-s + 0.850i·29^-s + 0.699·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.306i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.951 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$0.951 + 0.306i$
Analytic conductor:	$79.2985$
Root analytic conductor:	$8.90497$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1344} (223, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1344,\ (\ :3/2),\ 0.951 + 0.306i)$

Particular Values

$L(2)$	$\approx$	$3.285129603$
$L(\frac12)$	$\approx$	$3.285129603$
$L(\frac{5}{2})$		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 - 3iT $
	7	$ 1 + (-16.4 + 8.44i)T $
good	5	$ 1 - 16.7T + 125T^{2} $
	11	$ 1 + 8.98T + 1.33e3T^{2} $
	13	$ 1 - 6.28T + 2.19e3T^{2} $
	17	$ 1 + 106. iT - 4.91e3T^{2} $
	19	$ 1 + 25.1iT - 6.85e3T^{2} $
	23	$ 1 - 19.5iT - 1.21e4T^{2} $
	29	$ 1 - 132. iT - 2.43e4T^{2} $
	31	$ 1 - 120.T + 2.97e4T^{2} $
	37	$ 1 + 72.1iT - 5.06e4T^{2} $
	41	$ 1 + 410. iT - 6.89e4T^{2} $
	43	$ 1 - 64.1T + 7.95e4T^{2} $
	47	$ 1 - 281.T + 1.03e5T^{2} $
	53	$ 1 + 401. iT - 1.48e5T^{2} $
	59	$ 1 - 584. iT - 2.05e5T^{2} $
	61	$ 1 - 736.T + 2.26e5T^{2} $
	67	$ 1 + 643.T + 3.00e5T^{2} $
	71	$ 1 + 186. iT - 3.57e5T^{2} $
	73	$ 1 + 64.5iT - 3.89e5T^{2} $
	79	$ 1 + 50.5iT - 4.93e5T^{2} $
	83	$ 1 - 17.9iT - 5.71e5T^{2} $
	89	$ 1 + 378. iT - 7.04e5T^{2} $
	97	$ 1 - 593. iT - 9.12e5T^{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

−9.229141906173702565774109537912, −8.680278570473544726060480746780, −7.50386071639604202596526760348, −6.74956595146467907008367895692, −5.54673781707451636377424453056, −5.18299639922832185880115671877, −4.21239579418140887680637381205, −2.87971796952900668824693791285, −1.98689127040353478798955351166, −0.78257310365290586050710929506, 1.19739236419564440609516387075, 1.93386929724015038100095988291, 2.73048271960889662585222696259, 4.27269488053621682787780042801, 5.35974740535782234641823689767, 5.98852655797351868154018751988, 6.56699285529137748708126683522, 7.87001325352835998447145947482, 8.368721039323275379636112586606, 9.250622920099967723420510148549

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(3.285129603\)
\(L(\frac12)\)	\(\approx\)	\(3.285129603\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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