LMFDB - L-function 2-28e2-28.27-c3-0-54

Dirichlet series

L(s) = 1

+ 9.29·3^-s − 19.8i·5^-s + 59.3·9^-s − 11.8i·11^-s − 19.9i·13^-s − 184. i·15^-s − 1.81i·17^-s − 7.32·19^-s − 106. i·23^-s − 269.·25^-s + 300.·27^-s − 191.·29^-s + 125.·31^-s − 110. i·33^-s + 316.·37^-s + ⋯

L(s) = 1

+ 1.78·3^-s − 1.77i·5^-s + 2.19·9^-s − 0.324i·11^-s − 0.426i·13^-s − 3.17i·15^-s − 0.0259i·17^-s − 0.0884·19^-s − 0.965i·23^-s − 2.15·25^-s + 2.14·27^-s − 1.22·29^-s + 0.725·31^-s − 0.580i·33^-s + 1.40·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 + 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.188 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$784$ = $2^{4} \cdot 7^{2}$
Sign:	$-0.188 + 0.981i$
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:	$(2,\ 784,\ (\ :3/2),\ -0.188 + 0.981i)$

Particular Values

$L(2)$	$\approx$	$3.867586927$
$L(\frac12)$	$\approx$	$3.867586927$
$L(\frac{5}{2})$		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 - 9.29T + 27T^{2} $
	5	$ 1 + 19.8iT - 125T^{2} $
	11	$ 1 + 11.8iT - 1.33e3T^{2} $
	13	$ 1 + 19.9iT - 2.19e3T^{2} $
	17	$ 1 + 1.81iT - 4.91e3T^{2} $
	19	$ 1 + 7.32T + 6.85e3T^{2} $
	23	$ 1 + 106. iT - 1.21e4T^{2} $
	29	$ 1 + 191.T + 2.43e4T^{2} $
	31	$ 1 - 125.T + 2.97e4T^{2} $
	37	$ 1 - 316.T + 5.06e4T^{2} $
	41	$ 1 - 321. iT - 6.89e4T^{2} $
	43	$ 1 - 74.3iT - 7.95e4T^{2} $
	47	$ 1 - 154.T + 1.03e5T^{2} $
	53	$ 1 + 319.T + 1.48e5T^{2} $
	59	$ 1 + 61.1T + 2.05e5T^{2} $
	61	$ 1 + 309. iT - 2.26e5T^{2} $
	67	$ 1 + 594. iT - 3.00e5T^{2} $
	71	$ 1 - 48.6iT - 3.57e5T^{2} $
	73	$ 1 - 770. iT - 3.89e5T^{2} $
	79	$ 1 + 960. iT - 4.93e5T^{2} $
	83	$ 1 + 1.06e3T + 5.71e5T^{2} $
	89	$ 1 + 655. iT - 7.04e5T^{2} $
	97	$ 1 - 704. 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

−9.388122744591036669348227131496, −8.756114923408636284527910256390, −8.155187562297877230997048060623, −7.60039533118947630262360284274, −6.11181591370918942337768561138, −4.82371583520234356623002174211, −4.15345458324542220162064044536, −3.03451518560341519589653225176, −1.88380955090804329584445160788, −0.797229490541430723403999622010, 1.85526056471910976967739100369, 2.63867766832856891560159701413, 3.44678324066933687507606838351, 4.21100796312285958432829975124, 5.98178008818896197147017999489, 7.16543393724495529754037411851, 7.39153013089365495691544094863, 8.378515790998417536232837248700, 9.419066344233663881847689179120, 9.893873246341729928952979661753

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(3.867586927\)
\(L(\frac12)\)	\(\approx\)	\(3.867586927\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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