LMFDB - L-function 2-2e8-8.5-c7-0-14

Dirichlet series

L(s) = 1

+ 2.95i·3^-s − 362. i·5^-s − 715.·7^-s + 2.17e3·9^-s + 7.29e3i·11^-s − 1.03e4i·13^-s + 1.07e3·15^-s − 1.12e4·17^-s + 2.66e4i·19^-s − 2.11e3i·21^-s − 2.35e4·23^-s − 5.32e4·25^-s + 1.29e4i·27^-s + 4.41e4i·29^-s − 3.11e5·31^-s + ⋯

L(s) = 1

+ 0.0632i·3^-s − 1.29i·5^-s − 0.787·7^-s + 0.995·9^-s + 1.65i·11^-s − 1.31i·13^-s + 0.0820·15^-s − 0.554·17^-s + 0.892i·19^-s − 0.0498i·21^-s − 0.403·23^-s − 0.681·25^-s + 0.126i·27^-s + 0.336i·29^-s − 1.87·31^-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

$L(4)$	$\approx$	$1.484068650$
$L(\frac12)$	$\approx$	$1.484068650$
$L(\frac{9}{2})$		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 - 2.95iT - 2.18e3T^{2} $
	5	$ 1 + 362. iT - 7.81e4T^{2} $
	7	$ 1 + 715.T + 8.23e5T^{2} $
	11	$ 1 - 7.29e3iT - 1.94e7T^{2} $
	13	$ 1 + 1.03e4iT - 6.27e7T^{2} $
	17	$ 1 + 1.12e4T + 4.10e8T^{2} $
	19	$ 1 - 2.66e4iT - 8.93e8T^{2} $
	23	$ 1 + 2.35e4T + 3.40e9T^{2} $
	29	$ 1 - 4.41e4iT - 1.72e10T^{2} $
	31	$ 1 + 3.11e5T + 2.75e10T^{2} $
	37	$ 1 - 1.39e5iT - 9.49e10T^{2} $
	41	$ 1 - 7.53e5T + 1.94e11T^{2} $
	43	$ 1 + 1.25e5iT - 2.71e11T^{2} $
	47	$ 1 - 7.98e5T + 5.06e11T^{2} $
	53	$ 1 - 1.14e6iT - 1.17e12T^{2} $
	59	$ 1 + 2.01e6iT - 2.48e12T^{2} $
	61	$ 1 - 5.67e3iT - 3.14e12T^{2} $
	67	$ 1 - 3.80e6iT - 6.06e12T^{2} $
	71	$ 1 - 2.78e6T + 9.09e12T^{2} $
	73	$ 1 - 1.85e6T + 1.10e13T^{2} $
	79	$ 1 + 2.26e6T + 1.92e13T^{2} $
	83	$ 1 - 2.79e6iT - 2.71e13T^{2} $
	89	$ 1 - 5.86e6T + 4.42e13T^{2} $
	97	$ 1 + 1.46e7T + 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

−10.64745181684091181746563690542, −9.793805187300032222000606283807, −9.190393200226963461923613134738, −7.913498028282631166801907336757, −7.07893544534935079931713604909, −5.70366138466731868732678950976, −4.69657816004705118433897306615, −3.79294029951798148963095882851, −2.06294726467352233028406604128, −0.930324732779931117048550557065, 0.40946986239079721841197335021, 2.10540455298829751109933998421, 3.24807781964716934271518009678, 4.16978886227099154952922046237, 5.93710800083193659885683185791, 6.71163944275683312798233872365, 7.39269979395075542100092978391, 8.907154027641478993083200656625, 9.678699177465684839825289244051, 10.92660603136837245735461540624

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 $\F_{q}$
Belyi maps

\(L(4)\)	\(\approx\)	\(1.484068650\)
\(L(\frac12)\)	\(\approx\)	\(1.484068650\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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