LMFDB - L-function 2-48e2-8.3-c2-0-2

Dirichlet series

L(s) = 1

− 8.92i·5^-s + 10.9i·7^-s − 1.07·11^-s − 3.85i·13^-s − 7.85·17^-s + 17.0·19^-s − 8i·23^-s − 54.7·25^-s + 3.07i·29^-s + 30.6i·31^-s + 97.5·35^-s − 45.7i·37^-s − 35.8·41^-s − 74.6·43^-s + 42.1i·47^-s + ⋯

L(s) = 1

− 1.78i·5^-s + 1.56i·7^-s − 0.0974·11^-s − 0.296i·13^-s − 0.462·17^-s + 0.898·19^-s − 0.347i·23^-s − 2.18·25^-s + 0.105i·29^-s + 0.988i·31^-s + 2.78·35^-s − 1.23i·37^-s − 0.874·41^-s − 1.73·43^-s + 0.896i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2304$ = $2^{8} \cdot 3^{2}$
Sign:	$-0.707 - 0.707i$
Analytic conductor:	$62.7794$
Root analytic conductor:	$7.92334$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2304} (127, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2304,\ (\ :1),\ -0.707 - 0.707i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.2503201344$
$L(\frac12)$	$\approx$	$0.2503201344$
$L(2)$		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 + 8.92iT - 25T^{2} $
	7	$ 1 - 10.9iT - 49T^{2} $
	11	$ 1 + 1.07T + 121T^{2} $
	13	$ 1 + 3.85iT - 169T^{2} $
	17	$ 1 + 7.85T + 289T^{2} $
	19	$ 1 - 17.0T + 361T^{2} $
	23	$ 1 + 8iT - 529T^{2} $
	29	$ 1 - 3.07iT - 841T^{2} $
	31	$ 1 - 30.6iT - 961T^{2} $
	37	$ 1 + 45.7iT - 1.36e3T^{2} $
	41	$ 1 + 35.8T + 1.68e3T^{2} $
	43	$ 1 + 74.6T + 1.84e3T^{2} $
	47	$ 1 - 42.1iT - 2.20e3T^{2} $
	53	$ 1 - 12.9iT - 2.80e3T^{2} $
	59	$ 1 - 44.2T + 3.48e3T^{2} $
	61	$ 1 + 14iT - 3.72e3T^{2} $
	67	$ 1 + 80.4T + 4.48e3T^{2} $
	71	$ 1 + 123. iT - 5.04e3T^{2} $
	73	$ 1 - 85.4T + 5.32e3T^{2} $
	79	$ 1 - 55.2iT - 6.24e3T^{2} $
	83	$ 1 + 49.0T + 6.88e3T^{2} $
	89	$ 1 + 105.T + 7.92e3T^{2} $
	97	$ 1 - 21.1T + 9.40e3T^{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

−9.135410804461174701812511125349, −8.428374347524229243940091953252, −7.944779617447313049102138785555, −6.68912176244868830136731596931, −5.65071406152909454608340190516, −5.24390863110268170509398236268, −4.58233623463315519606493747368, −3.35184001112761191888020731096, −2.20914079603164346614666299011, −1.25757110853355560600988955333, 0.06039082645072214260553643347, 1.57986694202122553515310138406, 2.81482336110461661548826111365, 3.55532674674327245243394971705, 4.25969409547056672157630634455, 5.44451156104615134181413449917, 6.63759757735660733195634838865, 6.86520655100522140525482337241, 7.56446356148674986649542506666, 8.301333454379185971949797292982

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.2503201344\)
\(L(\frac12)\)	\(\approx\)	\(0.2503201344\)
\(L(2)\)		not available
\(L(1)\)		not available

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