LMFDB - L-function 2-5472-12.11-c1-0-66

Dirichlet series

L(s) = 1

− 3.62i·5^-s − 0.712i·7^-s + 3.62·11^-s − 3.02·13^-s − 5.59i·17^-s + i·19^-s + 2.37·23^-s − 8.14·25^-s − 1.44i·29^-s − 1.02i·31^-s − 2.58·35^-s − 3.02·37^-s − 2.01i·41^-s − 7.78i·43^-s − 6.97·47^-s + ⋯

L(s) = 1

− 1.62i·5^-s − 0.269i·7^-s + 1.09·11^-s − 0.838·13^-s − 1.35i·17^-s + 0.229i·19^-s + 0.494·23^-s − 1.62·25^-s − 0.269i·29^-s − 0.184i·31^-s − 0.436·35^-s − 0.497·37^-s − 0.314i·41^-s − 1.18i·43^-s − 1.01·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5472$ = $2^{5} \cdot 3^{2} \cdot 19$
Sign:	$-0.985 + 0.169i$
Analytic conductor:	$43.6941$
Root analytic conductor:	$6.61015$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5472} (2015, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5472,\ (\ :1/2),\ -0.985 + 0.169i)$

Particular Values

$L(1)$	$\approx$	$1.378382658$
$L(\frac12)$	$\approx$	$1.378382658$
$L(\frac{3}{2})$		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 - iT $
good	5	$ 1 + 3.62iT - 5T^{2} $
	7	$ 1 + 0.712iT - 7T^{2} $
	11	$ 1 - 3.62T + 11T^{2} $
	13	$ 1 + 3.02T + 13T^{2} $
	17	$ 1 + 5.59iT - 17T^{2} $
	23	$ 1 - 2.37T + 23T^{2} $
	29	$ 1 + 1.44iT - 29T^{2} $
	31	$ 1 + 1.02iT - 31T^{2} $
	37	$ 1 + 3.02T + 37T^{2} $
	41	$ 1 + 2.01iT - 41T^{2} $
	43	$ 1 + 7.78iT - 43T^{2} $
	47	$ 1 + 6.97T + 47T^{2} $
	53	$ 1 - 2.40iT - 53T^{2} $
	59	$ 1 - 5.80T + 59T^{2} $
	61	$ 1 + 11.8T + 61T^{2} $
	67	$ 1 + 5.42iT - 67T^{2} $
	71	$ 1 + 1.87T + 71T^{2} $
	73	$ 1 - 5.49T + 73T^{2} $
	79	$ 1 + 7.07iT - 79T^{2} $
	83	$ 1 + 5.20T + 83T^{2} $
	89	$ 1 - 2.68iT - 89T^{2} $
	97	$ 1 - 12.9T + 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

−7.78364566976832718256668838237, −7.23767874037490481201546784540, −6.40607160452122964914997604886, −5.43273683669805964951636492797, −4.89779814711489717014719195828, −4.29855349827477136790976879269, −3.44000971845736576625112785225, −2.21574021438263846990918801338, −1.22415884528600108742922121207, −0.37344898734763125636612561571, 1.49433871293300972041524742012, 2.44618063360398784136233647473, 3.21742880183010961216810680406, 3.89178258452401357394299674566, 4.81971289721668585395017394870, 5.87965359250983443731087714022, 6.45845187732756547406844047936, 6.97090326741199005832470931052, 7.62568889185941404492532894487, 8.475670327260018290657309861724

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(1.378382658\)
\(L(\frac12)\)	\(\approx\)	\(1.378382658\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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