LMFDB - L-function 2-72e2-24.11-c1-0-77

Dirichlet series

L(s) = 1

+ 2.24·5^-s − 3.77i·7^-s + 2.57i·11^-s + 2.77i·13^-s − 4.81i·17^-s + 5.77·19^-s + 1.91·23^-s + 0.0289·25^-s − 10.2·29^-s − 6.22i·31^-s − 8.45i·35^-s − 1.96i·37^-s − 10.6i·41^-s + 5.01·43^-s + 6.36·47^-s + ⋯

L(s) = 1

+ 1.00·5^-s − 1.42i·7^-s + 0.776i·11^-s + 0.768i·13^-s − 1.16i·17^-s + 1.32·19^-s + 0.398·23^-s + 0.00579·25^-s − 1.90·29^-s − 1.11i·31^-s − 1.42i·35^-s − 0.322i·37^-s − 1.65i·41^-s + 0.764·43^-s + 0.929·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5184$ = $2^{6} \cdot 3^{4}$
Sign:	$0.258 + 0.965i$
Analytic conductor:	$41.3944$
Root analytic conductor:	$6.43385$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5184} (2591, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5184,\ (\ :1/2),\ 0.258 + 0.965i)$

Particular Values

$L(1)$	$\approx$	$2.265908484$
$L(\frac12)$	$\approx$	$2.265908484$
$L(\frac{3}{2})$		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 - 2.24T + 5T^{2} $
	7	$ 1 + 3.77iT - 7T^{2} $
	11	$ 1 - 2.57iT - 11T^{2} $
	13	$ 1 - 2.77iT - 13T^{2} $
	17	$ 1 + 4.81iT - 17T^{2} $
	19	$ 1 - 5.77T + 19T^{2} $
	23	$ 1 - 1.91T + 23T^{2} $
	29	$ 1 + 10.2T + 29T^{2} $
	31	$ 1 + 6.22iT - 31T^{2} $
	37	$ 1 + 1.96iT - 37T^{2} $
	41	$ 1 + 10.6iT - 41T^{2} $
	43	$ 1 - 5.01T + 43T^{2} $
	47	$ 1 - 6.36T + 47T^{2} $
	53	$ 1 + 2.35T + 53T^{2} $
	59	$ 1 - 5.22iT - 59T^{2} $
	61	$ 1 - 2.26iT - 61T^{2} $
	67	$ 1 - 12.5T + 67T^{2} $
	71	$ 1 + 1.19T + 71T^{2} $
	73	$ 1 - 8.57T + 73T^{2} $
	79	$ 1 + 2.39iT - 79T^{2} $
	83	$ 1 + 15.3iT - 83T^{2} $
	89	$ 1 - 14.7iT - 89T^{2} $
	97	$ 1 + 13.7T + 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

−7.72841656224209055860420335236, −7.27004606808862851345838708751, −6.89458905152094572095791916044, −5.77160352726870200647473947333, −5.24481527095553021199311639697, −4.27894892850685182921871608673, −3.70697292045799174008098195994, −2.48883091569632697561217971387, −1.69655962746198192582876284979, −0.61856657291801898081342233182, 1.20648744714710260857257497019, 2.12707466117377212560692114721, 2.96123672720424490219840162322, 3.65887539163589673191303973775, 5.07629145094181065498438443517, 5.60761124931937282380820658696, 5.92113644924861056931735397225, 6.75008308171994111328916071773, 7.86275101575576694276619157231, 8.343778197515906800014968136626

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.265908484\)
\(L(\frac12)\)	\(\approx\)	\(2.265908484\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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