LMFDB - L-function 2-24e2-8.5-c5-0-28

Dirichlet series

L(s) = 1

+ 474i·11^-s − 1.91e3·17^-s − 2.88e3i·19^-s + 3.12e3·25^-s + 1.39e4·41^-s + 2.25e4i·43^-s − 1.68e4·49^-s − 4.84e4i·59^-s − 6.71e4i·67^-s + 5.04e4·73^-s + 8.92e4i·83^-s − 7.21e3·89^-s + 8.54e4·97^-s − 2.35e5i·107^-s + 2.56e5·113^-s + ⋯

L(s) = 1

+ 1.18i·11^-s − 1.60·17^-s − 1.83i·19^-s + 25^-s + 1.29·41^-s + 1.85i·43^-s − 49^-s − 1.81i·59^-s − 1.82i·67^-s + 1.10·73^-s + 1.42i·83^-s − 0.0965·89^-s + 0.922·97^-s − 1.99i·107^-s + 1.88·113^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$576$ = $2^{6} \cdot 3^{2}$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$92.3810$
Root analytic conductor:	$9.61150$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{576} (289, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 576,\ (\ :5/2),\ 0.707 + 0.707i)$

Particular Values

$L(3)$	$\approx$	$1.647840128$
$L(\frac12)$	$\approx$	$1.647840128$
$L(\frac{7}{2})$		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 $
	3	$ 1 $
good	5	$ 1 - 3.12e3T^{2} $
	7	$ 1 + 1.68e4T^{2} $
	11	$ 1 - 474iT - 1.61e5T^{2} $
	13	$ 1 - 3.71e5T^{2} $
	17	$ 1 + 1.91e3T + 1.41e6T^{2} $
	19	$ 1 + 2.88e3iT - 2.47e6T^{2} $
	23	$ 1 + 6.43e6T^{2} $
	29	$ 1 - 2.05e7T^{2} $
	31	$ 1 + 2.86e7T^{2} $
	37	$ 1 - 6.93e7T^{2} $
	41	$ 1 - 1.39e4T + 1.15e8T^{2} $
	43	$ 1 - 2.25e4iT - 1.47e8T^{2} $
	47	$ 1 + 2.29e8T^{2} $
	53	$ 1 - 4.18e8T^{2} $
	59	$ 1 + 4.84e4iT - 7.14e8T^{2} $
	61	$ 1 - 8.44e8T^{2} $
	67	$ 1 + 6.71e4iT - 1.35e9T^{2} $
	71	$ 1 + 1.80e9T^{2} $
	73	$ 1 - 5.04e4T + 2.07e9T^{2} $
	79	$ 1 + 3.07e9T^{2} $
	83	$ 1 - 8.92e4iT - 3.93e9T^{2} $
	89	$ 1 + 7.21e3T + 5.58e9T^{2} $
	97	$ 1 - 8.54e4T + 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

−9.620996767315007904391817734179, −9.141346741509120794417804653929, −8.062053743725241159143843129563, −6.98310914501954769083002476773, −6.46722335784339154647228188882, −4.90078512574374387183288917465, −4.45639616117755830161647138285, −2.90274307540926298276840629271, −1.94851529069146397671304478108, −0.46534604674158695228640039938, 0.830841563478261781526188252394, 2.16262725381901011422446842724, 3.38373989225849911340225673098, 4.35209774834034559752032163164, 5.59759594426830401548787868606, 6.33960870585120350772318373908, 7.38815623710341371733546409396, 8.470355453335215225436345369853, 8.962622542544598396955652352631, 10.19246956796602388367416061901

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(1.647840128\)
\(L(\frac12)\)	\(\approx\)	\(1.647840128\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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