LMFDB - L-function 2-2736-57.56-c1-0-30

Dirichlet series

L(s) = 1

− 2.60i·5^-s + 0.723·7^-s − 5.10i·11^-s + 5.91i·13^-s − 2.27i·17^-s + (4.22 − 1.08i)19^-s − 0.0606i·23^-s − 1.80·25^-s − 2.46·29^-s − 8.86i·31^-s − 1.88i·35^-s + 1.03i·37^-s + 10.9·41^-s − 0.413·43^-s − 1.94i·47^-s + ⋯

L(s) = 1

− 1.16i·5^-s + 0.273·7^-s − 1.53i·11^-s + 1.63i·13^-s − 0.552i·17^-s + (0.968 − 0.248i)19^-s − 0.0126i·23^-s − 0.361·25^-s − 0.458·29^-s − 1.59i·31^-s − 0.318i·35^-s + 0.170i·37^-s + 1.71·41^-s − 0.0630·43^-s − 0.284i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$-0.356 + 0.934i$
Analytic conductor:	$21.8470$
Root analytic conductor:	$4.67408$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2736} (1025, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2736,\ (\ :1/2),\ -0.356 + 0.934i)$

Particular Values

$L(1)$	$\approx$	$1.646388631$
$L(\frac12)$	$\approx$	$1.646388631$
$L(\frac{3}{2})$		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 $
	19	$ 1 + (-4.22 + 1.08i)T $
good	5	$ 1 + 2.60iT - 5T^{2} $
	7	$ 1 - 0.723T + 7T^{2} $
	11	$ 1 + 5.10iT - 11T^{2} $
	13	$ 1 - 5.91iT - 13T^{2} $
	17	$ 1 + 2.27iT - 17T^{2} $
	23	$ 1 + 0.0606iT - 23T^{2} $
	29	$ 1 + 2.46T + 29T^{2} $
	31	$ 1 + 8.86iT - 31T^{2} $
	37	$ 1 - 1.03iT - 37T^{2} $
	41	$ 1 - 10.9T + 41T^{2} $
	43	$ 1 + 0.413T + 43T^{2} $
	47	$ 1 + 1.94iT - 47T^{2} $
	53	$ 1 - 6.82T + 53T^{2} $
	59	$ 1 + 5.46T + 59T^{2} $
	61	$ 1 - 7.54T + 61T^{2} $
	67	$ 1 + 12.6iT - 67T^{2} $
	71	$ 1 + 12.3T + 71T^{2} $
	73	$ 1 + 14.3T + 73T^{2} $
	79	$ 1 - 9.40iT - 79T^{2} $
	83	$ 1 + 8.01iT - 83T^{2} $
	89	$ 1 + 16.2T + 89T^{2} $
	97	$ 1 - 0.783iT - 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

−8.754313792399988979155399647727, −7.937944875518450300068036251108, −7.17065965121395023902392369401, −6.12333231125127171208574765142, −5.50332998712869532108213207037, −4.62247283615296470662634236219, −3.96974611539572006380423912706, −2.81526158777226776258016052358, −1.57110586327258202307608169211, −0.56368098727671031124915985468, 1.38234671461081586838014478821, 2.59368658166077178665789697429, 3.25411463975647464239616573939, 4.28866795992003026977326656880, 5.27190622165687919507784633805, 5.95688045147910849206435426840, 7.03011882238547498762255126995, 7.40314344648813498695798172651, 8.105918035875025405422072866592, 9.108715275463358567489996580373

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.646388631\)
\(L(\frac12)\)	\(\approx\)	\(1.646388631\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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