LMFDB - L-function 2-2e8-8.5-c7-0-4

Dirichlet series

L(s) = 1

− 84i·3^-s + 82i·5^-s + 456·7^-s − 4.86e3·9^-s + 2.52e3i·11^-s − 1.07e4i·13^-s + 6.88e3·15^-s − 1.11e4·17^-s + 4.12e3i·19^-s − 3.83e4i·21^-s − 8.17e4·23^-s + 7.14e4·25^-s + 2.25e5i·27^-s + 9.97e4i·29^-s − 4.04e4·31^-s + ⋯

L(s) = 1

− 1.79i·3^-s + 0.293i·5^-s + 0.502·7^-s − 2.22·9^-s + 0.571i·11^-s − 1.36i·13^-s + 0.526·15^-s − 0.550·17^-s + 0.137i·19^-s − 0.902i·21^-s − 1.40·23^-s + 0.913·25^-s + 2.20i·27^-s + 0.759i·29^-s − 0.244·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$256$ = $2^{8}$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$79.9705$
Root analytic conductor:	$8.94262$
Motivic weight:	$7$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{256} (129, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 256,\ (\ :7/2),\ 0.707 - 0.707i)$

Particular Values

$L(4)$	$\approx$	$0.6700756648$
$L(\frac12)$	$\approx$	$0.6700756648$
$L(\frac{9}{2})$		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 $
good	3	$ 1 + 84iT - 2.18e3T^{2} $
	5	$ 1 - 82iT - 7.81e4T^{2} $
	7	$ 1 - 456T + 8.23e5T^{2} $
	11	$ 1 - 2.52e3iT - 1.94e7T^{2} $
	13	$ 1 + 1.07e4iT - 6.27e7T^{2} $
	17	$ 1 + 1.11e4T + 4.10e8T^{2} $
	19	$ 1 - 4.12e3iT - 8.93e8T^{2} $
	23	$ 1 + 8.17e4T + 3.40e9T^{2} $
	29	$ 1 - 9.97e4iT - 1.72e10T^{2} $
	31	$ 1 + 4.04e4T + 2.75e10T^{2} $
	37	$ 1 - 4.19e5iT - 9.49e10T^{2} $
	41	$ 1 + 1.41e5T + 1.94e11T^{2} $
	43	$ 1 - 6.90e5iT - 2.71e11T^{2} $
	47	$ 1 + 6.82e5T + 5.06e11T^{2} $
	53	$ 1 + 1.81e6iT - 1.17e12T^{2} $
	59	$ 1 - 9.66e5iT - 2.48e12T^{2} $
	61	$ 1 - 1.88e6iT - 3.14e12T^{2} $
	67	$ 1 - 2.96e6iT - 6.06e12T^{2} $
	71	$ 1 - 2.54e6T + 9.09e12T^{2} $
	73	$ 1 - 1.68e6T + 1.10e13T^{2} $
	79	$ 1 - 4.03e6T + 1.92e13T^{2} $
	83	$ 1 + 5.38e6iT - 2.71e13T^{2} $
	89	$ 1 - 6.47e6T + 4.42e13T^{2} $
	97	$ 1 + 6.06e6T + 8.07e13T^{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

−11.14925570086343818736714994976, −10.04218243679370980611432845593, −8.453648364449749936029808142988, −7.921895020352833980086120407616, −6.98445456768812272635542563173, −6.17195982811732043725960572218, −5.01206895375848133957518027066, −3.12577401834294823773825107298, −2.05363110490017659800337975830, −1.09218341792945546554962518828, 0.16054144682128735812978866171, 2.12233151472190567563808185408, 3.64416350855205688470013100548, 4.40262392439682416425100183962, 5.24091705598414179955281739247, 6.40310702394613329366114798368, 8.105087292105323844963876075342, 9.004117622771594297649082735016, 9.577656471168676227115989819965, 10.72532125753540279067780307607

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(0.6700756648\)
\(L(\frac12)\)	\(\approx\)	\(0.6700756648\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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