LMFDB - L-function 2-28e2-28.27-c3-0-50

Dirichlet series

L(s) = 1

+ 9.98·3^-s + 9.37i·5^-s + 72.6·9^-s − 44.5i·11^-s − 71.8i·13^-s + 93.5i·15^-s − 52.6i·17^-s + 68.8·19^-s − 155. i·23^-s + 37.2·25^-s + 456.·27^-s − 131.·29^-s + 2.00·31^-s − 444. i·33^-s − 277.·37^-s + ⋯

L(s) = 1

+ 1.92·3^-s + 0.838i·5^-s + 2.69·9^-s − 1.22i·11^-s − 1.53i·13^-s + 1.61i·15^-s − 0.750i·17^-s + 0.831·19^-s − 1.41i·23^-s + 0.297·25^-s + 3.25·27^-s − 0.843·29^-s + 0.0116·31^-s − 2.34i·33^-s − 1.23·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$784$ = $2^{4} \cdot 7^{2}$
Sign:	$0.912 + 0.409i$
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:	$(2,\ 784,\ (\ :3/2),\ 0.912 + 0.409i)$

Particular Values

$L(2)$	$\approx$	$4.476722053$
$L(\frac12)$	$\approx$	$4.476722053$
$L(\frac{5}{2})$		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 - 9.98T + 27T^{2} $
	5	$ 1 - 9.37iT - 125T^{2} $
	11	$ 1 + 44.5iT - 1.33e3T^{2} $
	13	$ 1 + 71.8iT - 2.19e3T^{2} $
	17	$ 1 + 52.6iT - 4.91e3T^{2} $
	19	$ 1 - 68.8T + 6.85e3T^{2} $
	23	$ 1 + 155. iT - 1.21e4T^{2} $
	29	$ 1 + 131.T + 2.43e4T^{2} $
	31	$ 1 - 2.00T + 2.97e4T^{2} $
	37	$ 1 + 277.T + 5.06e4T^{2} $
	41	$ 1 - 272. iT - 6.89e4T^{2} $
	43	$ 1 - 445. iT - 7.95e4T^{2} $
	47	$ 1 - 135.T + 1.03e5T^{2} $
	53	$ 1 - 362.T + 1.48e5T^{2} $
	59	$ 1 - 488.T + 2.05e5T^{2} $
	61	$ 1 - 113. iT - 2.26e5T^{2} $
	67	$ 1 - 463. iT - 3.00e5T^{2} $
	71	$ 1 - 480. iT - 3.57e5T^{2} $
	73	$ 1 + 146. iT - 3.89e5T^{2} $
	79	$ 1 - 1.10e3iT - 4.93e5T^{2} $
	83	$ 1 + 727.T + 5.71e5T^{2} $
	89	$ 1 + 51.3iT - 7.04e5T^{2} $
	97	$ 1 + 556. 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

−9.811069426982796255085975829077, −8.800444112081129017477158128101, −8.226431160879578558505359667893, −7.48289603259024803612418538031, −6.67450147706750813439741149489, −5.33066715931933187947247772719, −3.91021696746004479386182661227, −2.92427759923097460019657038917, −2.75384699794676937459383217588, −0.993859496749190770960423728518, 1.56647334116511102702955655867, 2.06289762262752062557323513570, 3.57761050004487995993323178080, 4.19021471464703199751765716150, 5.22394390799017859388975008511, 7.02750292489968591086787999124, 7.38797773325007213475033290681, 8.476766338146195774984210073717, 9.119220651573256048863281632178, 9.523124332327752029918155367600

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(4.476722053\)
\(L(\frac12)\)	\(\approx\)	\(4.476722053\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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