LMFDB - L-function 2-28e2-28.27-c3-0-53

Dirichlet series

L(s) = 1

− 2.64·3^-s − 8.66i·5^-s − 20·9^-s − 68.7i·11^-s − 51.9i·13^-s + 22.9i·15^-s + 8.66i·17^-s − 66.1·19^-s + 50.4i·23^-s + 49.9·25^-s + 124.·27^-s − 168·29^-s + 224.·31^-s + 181. i·33^-s − 245·37^-s + ⋯

L(s) = 1

− 0.509·3^-s − 0.774i·5^-s − 0.740·9^-s − 1.88i·11^-s − 1.10i·13^-s + 0.394i·15^-s + 0.123i·17^-s − 0.798·19^-s + 0.456i·23^-s + 0.399·25^-s + 0.886·27^-s − 1.07·29^-s + 1.30·31^-s + 0.959i·33^-s − 1.08·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$784$ = $2^{4} \cdot 7^{2}$
Sign:	$-0.755 - 0.654i$
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:	$(2,\ 784,\ (\ :3/2),\ -0.755 - 0.654i)$

Particular Values

$L(2)$	$\approx$	$0.4084240536$
$L(\frac12)$	$\approx$	$0.4084240536$
$L(\frac{5}{2})$		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 + 2.64T + 27T^{2} $
	5	$ 1 + 8.66iT - 125T^{2} $
	11	$ 1 + 68.7iT - 1.33e3T^{2} $
	13	$ 1 + 51.9iT - 2.19e3T^{2} $
	17	$ 1 - 8.66iT - 4.91e3T^{2} $
	19	$ 1 + 66.1T + 6.85e3T^{2} $
	23	$ 1 - 50.4iT - 1.21e4T^{2} $
	29	$ 1 + 168T + 2.43e4T^{2} $
	31	$ 1 - 224.T + 2.97e4T^{2} $
	37	$ 1 + 245T + 5.06e4T^{2} $
	41	$ 1 + 45.0iT - 6.89e4T^{2} $
	43	$ 1 + 302. iT - 7.95e4T^{2} $
	47	$ 1 - 420.T + 1.03e5T^{2} $
	53	$ 1 + 345T + 1.48e5T^{2} $
	59	$ 1 + 436.T + 2.05e5T^{2} $
	61	$ 1 - 396. iT - 2.26e5T^{2} $
	67	$ 1 + 444. iT - 3.00e5T^{2} $
	71	$ 1 + 45.8iT - 3.57e5T^{2} $
	73	$ 1 - 961. iT - 3.89e5T^{2} $
	79	$ 1 + 206. iT - 4.93e5T^{2} $
	83	$ 1 - 888.T + 5.71e5T^{2} $
	89	$ 1 - 1.51e3iT - 7.04e5T^{2} $
	97	$ 1 + 433. 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

−9.145078700130522858455436975896, −8.538387784495069937397766595609, −7.931878925746541638941218171641, −6.48875049079608344879866455486, −5.66482545383489410152100080787, −5.19416789218002778290991279052, −3.77444112539174264425937772815, −2.77873542844993789577804383348, −0.998535940301535425862912308542, −0.13654086369084919078606318422, 1.81160431336031722212983258463, 2.79657990711086001486569936346, 4.23996692814771860896622475528, 4.97671711636998636680170809910, 6.29925035623360855122718749132, 6.78810965768654259919402317581, 7.66924655030202625263219299513, 8.840539188841802522279444205675, 9.658519386911562714971921747331, 10.52995725371630951503872455985

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.4084240536\)
\(L(\frac12)\)	\(\approx\)	\(0.4084240536\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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