LMFDB - L-function 2-1344-56.27-c3-0-23

Dirichlet series

L(s) = 1

− 3i·3^-s − 3.15·5^-s + (−6.11 − 17.4i)7^-s − 9·9^-s − 60.9·11^-s − 59.6·13^-s + 9.46i·15^-s − 21.4i·17^-s + 95.5i·19^-s + (−52.4 + 18.3i)21^-s + 46.3i·23^-s − 115.·25^-s + 27i·27^-s + 107. i·29^-s + 94.2·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 0.282·5^-s + (−0.329 − 0.944i)7^-s − 0.333·9^-s − 1.67·11^-s − 1.27·13^-s + 0.162i·15^-s − 0.305i·17^-s + 1.15i·19^-s + (−0.545 + 0.190i)21^-s + 0.419i·23^-s − 0.920·25^-s + 0.192i·27^-s + 0.688i·29^-s + 0.546·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0743i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$0.997 - 0.0743i$
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:	$(2,\ 1344,\ (\ :3/2),\ 0.997 - 0.0743i)$

Particular Values

$L(2)$	$\approx$	$0.7185280955$
$L(\frac12)$	$\approx$	$0.7185280955$
$L(\frac{5}{2})$		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 + (6.11 + 17.4i)T $
good	5	$ 1 + 3.15T + 125T^{2} $
	11	$ 1 + 60.9T + 1.33e3T^{2} $
	13	$ 1 + 59.6T + 2.19e3T^{2} $
	17	$ 1 + 21.4iT - 4.91e3T^{2} $
	19	$ 1 - 95.5iT - 6.85e3T^{2} $
	23	$ 1 - 46.3iT - 1.21e4T^{2} $
	29	$ 1 - 107. iT - 2.43e4T^{2} $
	31	$ 1 - 94.2T + 2.97e4T^{2} $
	37	$ 1 + 131. iT - 5.06e4T^{2} $
	41	$ 1 + 283. iT - 6.89e4T^{2} $
	43	$ 1 - 373.T + 7.95e4T^{2} $
	47	$ 1 - 136.T + 1.03e5T^{2} $
	53	$ 1 - 298. iT - 1.48e5T^{2} $
	59	$ 1 + 468. iT - 2.05e5T^{2} $
	61	$ 1 - 563.T + 2.26e5T^{2} $
	67	$ 1 + 160.T + 3.00e5T^{2} $
	71	$ 1 - 409. iT - 3.57e5T^{2} $
	73	$ 1 + 930. iT - 3.89e5T^{2} $
	79	$ 1 + 442. iT - 4.93e5T^{2} $
	83	$ 1 - 190. iT - 5.71e5T^{2} $
	89	$ 1 + 829. iT - 7.04e5T^{2} $
	97	$ 1 + 1.03e3iT - 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

−9.334857486888488389401409724911, −8.092697242488144206846645531231, −7.56217191001870978439395762912, −7.13306707066761692973575633339, −5.89884272105022249005592723974, −5.13167767552702322423862683940, −4.06776851841886166727867797732, −2.99888122541283663838912459228, −2.03354863044959269555002218655, −0.55616295231438879635817211879, 0.28491435450897338326160332524, 2.47744776223618316059645582918, 2.75057489543255403495297896367, 4.21960036183887712320987492482, 5.06762253566244403080727899635, 5.67207684592268708960876723719, 6.76902023667995487668752589473, 7.81351023787433932420128952982, 8.358382564430911749758243502467, 9.405211260346677428756936758715

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.7185280955\)
\(L(\frac12)\)	\(\approx\)	\(0.7185280955\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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