LMFDB - L-function 2-48e2-24.11-c1-0-2

Dirichlet series

L(s) = 1

− 4.24·5^-s − 4i·13^-s − 4.24i·17^-s + 12.9·25^-s − 4.24·29^-s + 2i·37^-s + 12.7i·41^-s + 7·49^-s − 12.7·53^-s + 10i·61^-s + 16.9i·65^-s − 16·73^-s + 17.9i·85^-s − 4.24i·89^-s − 8·97^-s + ⋯

L(s) = 1

− 1.89·5^-s − 1.10i·13^-s − 1.02i·17^-s + 2.59·25^-s − 0.787·29^-s + 0.328i·37^-s + 1.98i·41^-s + 49^-s − 1.74·53^-s + 1.28i·61^-s + 2.10i·65^-s − 1.87·73^-s + 1.95i·85^-s − 0.449i·89^-s − 0.812·97^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2304$ = $2^{8} \cdot 3^{2}$
Sign:	$-0.169 - 0.985i$
Analytic conductor:	$18.3975$
Root analytic conductor:	$4.28923$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2304} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2304,\ (\ :1/2),\ -0.169 - 0.985i)$

Particular Values

$L(1)$	$\approx$	$0.4596298777$
$L(\frac12)$	$\approx$	$0.4596298777$
$L(\frac{3}{2})$		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 + 4.24T + 5T^{2} $
	7	$ 1 - 7T^{2} $
	11	$ 1 - 11T^{2} $
	13	$ 1 + 4iT - 13T^{2} $
	17	$ 1 + 4.24iT - 17T^{2} $
	19	$ 1 + 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 + 4.24T + 29T^{2} $
	31	$ 1 - 31T^{2} $
	37	$ 1 - 2iT - 37T^{2} $
	41	$ 1 - 12.7iT - 41T^{2} $
	43	$ 1 + 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 + 12.7T + 53T^{2} $
	59	$ 1 - 59T^{2} $
	61	$ 1 - 10iT - 61T^{2} $
	67	$ 1 + 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 16T + 73T^{2} $
	79	$ 1 - 79T^{2} $
	83	$ 1 - 83T^{2} $
	89	$ 1 + 4.24iT - 89T^{2} $
	97	$ 1 + 8T + 97T^{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

−9.038534627588349294488918633682, −8.300186370334537605916365135142, −7.62741826018454340847434653563, −7.25116425811383875342943976422, −6.15427154776584537075707952763, −5.04255642342924243220267708301, −4.40398633882449808527785892196, −3.43198511704205012337449992451, −2.82908884832523445124157183624, −0.945711542954576152791818881692, 0.20421703015655479918572033501, 1.78661411746281092730120268500, 3.21554199202807981722130805072, 4.00392677020688365232810016452, 4.45228191969632204645319055120, 5.61225846447941715209799195022, 6.71444857011733650095223593529, 7.30511759388901392751278950844, 8.008324460356857040922706490590, 8.682430848374074638658816859965

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.4596298777\)
\(L(\frac12)\)	\(\approx\)	\(0.4596298777\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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