LMFDB - L-function 2-28e2-28.27-c3-0-46

Dirichlet series

L(s) = 1

+ 8.86·3^-s − 7.47i·5^-s + 51.5·9^-s + 61.2i·11^-s − 12.0i·13^-s − 66.2i·15^-s − 100. i·17^-s + 73.6·19^-s − 96.8i·23^-s + 69.1·25^-s + 217.·27^-s + 284.·29^-s − 256.·31^-s + 542. i·33^-s + 408.·37^-s + ⋯

L(s) = 1

+ 1.70·3^-s − 0.668i·5^-s + 1.90·9^-s + 1.67i·11^-s − 0.257i·13^-s − 1.14i·15^-s − 1.43i·17^-s + 0.889·19^-s − 0.877i·23^-s + 0.552·25^-s + 1.54·27^-s + 1.82·29^-s − 1.48·31^-s + 2.86i·33^-s + 1.81·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$784$ = $2^{4} \cdot 7^{2}$
Sign:	$0.933 + 0.358i$
Analytic conductor:	$46.2574$
Root analytic conductor:	$6.80128$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{784} (783, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 784,\ (\ :3/2),\ 0.933 + 0.358i)$

Particular Values

$L(2)$	$\approx$	$4.184306324$
$L(\frac12)$	$\approx$	$4.184306324$
$L(\frac{5}{2})$		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 $
	7	$ 1 $
good	3	$ 1 - 8.86T + 27T^{2} $
	5	$ 1 + 7.47iT - 125T^{2} $
	11	$ 1 - 61.2iT - 1.33e3T^{2} $
	13	$ 1 + 12.0iT - 2.19e3T^{2} $
	17	$ 1 + 100. iT - 4.91e3T^{2} $
	19	$ 1 - 73.6T + 6.85e3T^{2} $
	23	$ 1 + 96.8iT - 1.21e4T^{2} $
	29	$ 1 - 284.T + 2.43e4T^{2} $
	31	$ 1 + 256.T + 2.97e4T^{2} $
	37	$ 1 - 408.T + 5.06e4T^{2} $
	41	$ 1 - 120. iT - 6.89e4T^{2} $
	43	$ 1 - 308. iT - 7.95e4T^{2} $
	47	$ 1 + 26.0T + 1.03e5T^{2} $
	53	$ 1 + 311.T + 1.48e5T^{2} $
	59	$ 1 - 444.T + 2.05e5T^{2} $
	61	$ 1 + 488. iT - 2.26e5T^{2} $
	67	$ 1 - 585. iT - 3.00e5T^{2} $
	71	$ 1 + 416. iT - 3.57e5T^{2} $
	73	$ 1 + 819. iT - 3.89e5T^{2} $
	79	$ 1 + 318. iT - 4.93e5T^{2} $
	83	$ 1 - 218.T + 5.71e5T^{2} $
	89	$ 1 + 19.4iT - 7.04e5T^{2} $
	97	$ 1 + 306. 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

−9.485533414722842687997460018680, −9.190350675635072180695876563803, −8.114664951784288398157967332792, −7.54435335546274405456446297234, −6.71283726817417102567273077285, −4.93453744903108940943715231221, −4.44143869616635818058039382824, −3.08596669164628790752648265881, −2.32626250644420088550406414735, −1.08014124294446454027031168066, 1.24053866681453803976166166449, 2.56972563662762986418941434081, 3.32990985825143778462761108178, 3.98603391234728517934569371317, 5.61062137439082604483092864369, 6.65122825404529539781212355184, 7.60706061278328061572131759429, 8.355950694281190972444169128140, 8.900976750631652161446585042437, 9.804173818428458809380224666086

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(4.184306324\)
\(L(\frac12)\)	\(\approx\)	\(4.184306324\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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