LMFDB - L-function 2-2e6-8.5-c23-0-19

Dirichlet series

L(s) = 1

− 3.62e5i·3^-s − 4.97e7i·5^-s + 7.73e9·7^-s − 3.69e10·9^-s + 9.29e11i·11^-s + 1.06e13i·13^-s − 1.79e13·15^-s − 2.44e14·17^-s − 8.89e14i·19^-s − 2.80e15i·21^-s − 8.02e14·23^-s + 9.44e15·25^-s − 2.07e16i·27^-s + 8.69e16i·29^-s − 6.31e16·31^-s + ⋯

L(s) = 1

− 1.18i·3^-s − 0.455i·5^-s + 1.47·7^-s − 0.392·9^-s + 0.982i·11^-s + 1.64i·13^-s − 0.537·15^-s − 1.72·17^-s − 1.75i·19^-s − 1.74i·21^-s − 0.175·23^-s + 0.792·25^-s − 0.716i·27^-s + 1.32i·29^-s − 0.446·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(24-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$64$ = $2^{6}$
Sign:	$0.965 + 0.258i$
Analytic conductor:	$214.530$
Root analytic conductor:	$14.6468$
Motivic weight:	$23$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{64} (33, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 64,\ (\ :23/2),\ 0.965 + 0.258i)$

Particular Values

$L(12)$	$\approx$	$2.556487596$
$L(\frac12)$	$\approx$	$2.556487596$
$L(\frac{25}{2})$		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 + 3.62e5iT - 9.41e10T^{2} $
	5	$ 1 + 4.97e7iT - 1.19e16T^{2} $
	7	$ 1 - 7.73e9T + 2.73e19T^{2} $
	11	$ 1 - 9.29e11iT - 8.95e23T^{2} $
	13	$ 1 - 1.06e13iT - 4.17e25T^{2} $
	17	$ 1 + 2.44e14T + 1.99e28T^{2} $
	19	$ 1 + 8.89e14iT - 2.57e29T^{2} $
	23	$ 1 + 8.02e14T + 2.08e31T^{2} $
	29	$ 1 - 8.69e16iT - 4.31e33T^{2} $
	31	$ 1 + 6.31e16T + 2.00e34T^{2} $
	37	$ 1 - 9.15e17iT - 1.17e36T^{2} $
	41	$ 1 - 2.73e18T + 1.24e37T^{2} $
	43	$ 1 + 7.42e18iT - 3.71e37T^{2} $
	47	$ 1 + 2.26e19T + 2.87e38T^{2} $
	53	$ 1 + 2.93e18iT - 4.55e39T^{2} $
	59	$ 1 - 4.03e19iT - 5.36e40T^{2} $
	61	$ 1 - 2.25e20iT - 1.15e41T^{2} $
	67	$ 1 - 7.92e20iT - 9.99e41T^{2} $
	71	$ 1 - 3.47e21T + 3.79e42T^{2} $
	73	$ 1 - 4.69e20T + 7.18e42T^{2} $
	79	$ 1 - 5.80e21T + 4.42e43T^{2} $
	83	$ 1 - 2.09e21iT - 1.37e44T^{2} $
	89	$ 1 + 3.89e22T + 6.85e44T^{2} $
	97	$ 1 - 3.55e22T + 4.96e45T^{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

−11.01266604373883913698582878977, −9.174466278711306157756996631341, −8.466350662122810650965934188935, −7.09570683633705664671281480535, −6.79272917996514012316518299758, −4.86621906929850857509921745848, −4.45951495388803011233845763117, −2.21793626390858248028775366436, −1.81433027628564288037369033376, −0.857418400700531752571983092969, 0.52586371239281499750474729527, 1.89631357069342688497828047470, 3.18130633609844586665147992860, 4.16380096682535369391239917442, 5.10161530017694173578689914378, 6.07841635495500821618796000206, 7.80948605759538274824992401200, 8.487999318399090305014326024013, 9.829628332210138190404045923899, 10.93248847272243668514472764056

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 $\F_{q}$

\(L(12)\)	\(\approx\)	\(2.556487596\)
\(L(\frac12)\)	\(\approx\)	\(2.556487596\)
\(L(\frac{25}{2})\)		not available
\(L(1)\)		not available

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