LMFDB - L-function 2-2e8-8.5-c5-0-35

Dirichlet series

L(s) = 1

− 20i·3^-s − 74i·5^-s + 24·7^-s − 157·9^-s + 124i·11^-s − 478i·13^-s − 1.48e3·15^-s − 1.19e3·17^-s − 3.04e3i·19^-s − 480i·21^-s − 184·23^-s − 2.35e3·25^-s − 1.72e3i·27^-s + 3.28e3i·29^-s − 5.72e3·31^-s + ⋯

L(s) = 1

− 1.28i·3^-s − 1.32i·5^-s + 0.185·7^-s − 0.646·9^-s + 0.308i·11^-s − 0.784i·13^-s − 1.69·15^-s − 1.00·17^-s − 1.93i·19^-s − 0.237i·21^-s − 0.0725·23^-s − 0.752·25^-s − 0.454i·27^-s + 0.724i·29^-s − 1.07·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$256$ = $2^{8}$
Sign:	$-0.707 - 0.707i$
Analytic conductor:	$41.0582$
Root analytic conductor:	$6.40767$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{256} (129, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 256,\ (\ :5/2),\ -0.707 - 0.707i)$

Particular Values

$L(3)$	$\approx$	$1.289076427$
$L(\frac12)$	$\approx$	$1.289076427$
$L(\frac{7}{2})$		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 + 20iT - 243T^{2} $
	5	$ 1 + 74iT - 3.12e3T^{2} $
	7	$ 1 - 24T + 1.68e4T^{2} $
	11	$ 1 - 124iT - 1.61e5T^{2} $
	13	$ 1 + 478iT - 3.71e5T^{2} $
	17	$ 1 + 1.19e3T + 1.41e6T^{2} $
	19	$ 1 + 3.04e3iT - 2.47e6T^{2} $
	23	$ 1 + 184T + 6.43e6T^{2} $
	29	$ 1 - 3.28e3iT - 2.05e7T^{2} $
	31	$ 1 + 5.72e3T + 2.86e7T^{2} $
	37	$ 1 - 1.03e4iT - 6.93e7T^{2} $
	41	$ 1 - 8.88e3T + 1.15e8T^{2} $
	43	$ 1 + 9.18e3iT - 1.47e8T^{2} $
	47	$ 1 - 2.36e4T + 2.29e8T^{2} $
	53	$ 1 - 1.16e4iT - 4.18e8T^{2} $
	59	$ 1 - 1.68e4iT - 7.14e8T^{2} $
	61	$ 1 - 1.84e4iT - 8.44e8T^{2} $
	67	$ 1 - 1.55e4iT - 1.35e9T^{2} $
	71	$ 1 - 3.19e4T + 1.80e9T^{2} $
	73	$ 1 - 4.88e3T + 2.07e9T^{2} $
	79	$ 1 - 4.45e4T + 3.07e9T^{2} $
	83	$ 1 + 6.73e4iT - 3.93e9T^{2} $
	89	$ 1 + 7.19e4T + 5.58e9T^{2} $
	97	$ 1 - 4.88e4T + 8.58e9T^{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

−10.75040589700776725337575406126, −9.229950268671409782653893734179, −8.589217787170878939399746916915, −7.55129662951040587956018011145, −6.72148416912916828270731567983, −5.40172816013318895153980493445, −4.45689683287032976796443187614, −2.51397833256581579649624963905, −1.29168142364925529930606681074, −0.37179498207568367946466172821, 2.11416570911757065959179987682, 3.53163498938008397447214813871, 4.23253079952235268324931161162, 5.67053095283426978958064181116, 6.69912533323737357094511685650, 7.86954116099159367580833754256, 9.169575027248295091103858572276, 9.935374803085159455557329184629, 10.87806424959077785702887401027, 11.20893585873722462824573429754

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(1.289076427\)
\(L(\frac12)\)	\(\approx\)	\(1.289076427\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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