LMFDB - L-function 2-28e2-28.27-c3-0-14

Dirichlet series

L(s) = 1

− 3.30·3^-s + 13.4i·5^-s − 16.0·9^-s − 28.0i·11^-s − 27.3i·13^-s − 44.4i·15^-s + 1.11i·17^-s + 150.·19^-s + 67.4i·23^-s − 56.2·25^-s + 142.·27^-s + 217.·29^-s + 42.9·31^-s + 92.6i·33^-s − 238.·37^-s + ⋯

L(s) = 1

− 0.635·3^-s + 1.20i·5^-s − 0.595·9^-s − 0.768i·11^-s − 0.583i·13^-s − 0.765i·15^-s + 0.0158i·17^-s + 1.81·19^-s + 0.611i·23^-s − 0.449·25^-s + 1.01·27^-s + 1.39·29^-s + 0.249·31^-s + 0.488i·33^-s − 1.05·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$784$ = $2^{4} \cdot 7^{2}$
Sign:	$-0.101 - 0.994i$
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:	$(2,\ 784,\ (\ :3/2),\ -0.101 - 0.994i)$

Particular Values

$L(2)$	$\approx$	$1.195369297$
$L(\frac12)$	$\approx$	$1.195369297$
$L(\frac{5}{2})$		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 + 3.30T + 27T^{2} $
	5	$ 1 - 13.4iT - 125T^{2} $
	11	$ 1 + 28.0iT - 1.33e3T^{2} $
	13	$ 1 + 27.3iT - 2.19e3T^{2} $
	17	$ 1 - 1.11iT - 4.91e3T^{2} $
	19	$ 1 - 150.T + 6.85e3T^{2} $
	23	$ 1 - 67.4iT - 1.21e4T^{2} $
	29	$ 1 - 217.T + 2.43e4T^{2} $
	31	$ 1 - 42.9T + 2.97e4T^{2} $
	37	$ 1 + 238.T + 5.06e4T^{2} $
	41	$ 1 + 21.4iT - 6.89e4T^{2} $
	43	$ 1 - 219. iT - 7.95e4T^{2} $
	47	$ 1 - 156.T + 1.03e5T^{2} $
	53	$ 1 + 666.T + 1.48e5T^{2} $
	59	$ 1 + 782.T + 2.05e5T^{2} $
	61	$ 1 + 112. iT - 2.26e5T^{2} $
	67	$ 1 - 1.02e3iT - 3.00e5T^{2} $
	71	$ 1 - 621. iT - 3.57e5T^{2} $
	73	$ 1 + 272. iT - 3.89e5T^{2} $
	79	$ 1 + 837. iT - 4.93e5T^{2} $
	83	$ 1 - 105.T + 5.71e5T^{2} $
	89	$ 1 - 1.00e3iT - 7.04e5T^{2} $
	97	$ 1 - 1.84e3iT - 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

−10.37877539572093891984247452714, −9.421317273497616900296555032019, −8.308656867056799335289179052983, −7.45869052609508116384196335353, −6.52249926548057489132532854343, −5.80970335047281151669944974835, −4.98149915015054039909057360192, −3.31781422657557256347033899952, −2.87289301808168935586196800564, −0.987110009628187300307129374456, 0.44248068462268389905450696164, 1.56103476969827148048492208686, 3.09362331658298741210054665643, 4.61810925884258986371570403737, 5.02294976350909961744188135098, 6.02740550547641920901864777078, 7.00771909179380010393862548363, 8.057580726772407313603483591913, 8.902980621341704213696348619325, 9.573111851392496580881052196475

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.195369297\)
\(L(\frac12)\)	\(\approx\)	\(1.195369297\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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