LMFDB - L-function 4-2368e2-1.1-c0e2-0-2

Dirichlet series

L(s) = 1

+ 2·5^-s − 2·7^-s + 9^-s − 2·13^-s + 2·23^-s + 2·25^-s − 2·29^-s − 4·35^-s + 2·37^-s − 2·43^-s + 2·45^-s − 2·47^-s + 49^-s + 2·53^-s − 2·63^-s − 4·65^-s + 2·71^-s + 2·83^-s + 4·91^-s + 2·103^-s − 2·113^-s + 4·115^-s − 2·117^-s + 121^-s + 2·125^-s + 127^-s + 131^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5607424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$4$
Conductor:	$5607424$ = $2^{12} \cdot 37^{2}$
Sign:	$1$
Analytic conductor:	$1.39661$
Root analytic conductor:	$1.08709$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(4,\ 5607424,\ (\ :0, 0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.387404483$
$L(\frac12)$	$\approx$	$1.387404483$
$L(1)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$\Gal(F_p)$	$F_p(T)$
bad	2		$ 1 $
	37	$C_1$	$ ( 1 - T )^{2} $
good	3	$C_2^2$	$ 1 - T^{2} + T^{4} $
	5	$C_1$$\times$$C_2$	$ ( 1 - T )^{2}( 1 + T^{2} ) $
	7	$C_2$	$ ( 1 + T + T^{2} )^{2} $
	11	$C_2^2$	$ 1 - T^{2} + T^{4} $
	13	$C_1$$\times$$C_2$	$ ( 1 + T )^{2}( 1 + T^{2} ) $
	17	$C_2^2$	$ 1 + T^{4} $
	19	$C_2^2$	$ 1 + T^{4} $
	23	$C_1$$\times$$C_2$	$ ( 1 - T )^{2}( 1 + T^{2} ) $
	29	$C_1$$\times$$C_2$	$ ( 1 + T )^{2}( 1 + T^{2} ) $
	31	$C_2^2$	$ 1 + T^{4} $
	41	$C_2^2$	$ 1 - T^{2} + T^{4} $
	43	$C_1$$\times$$C_2$	$ ( 1 + T )^{2}( 1 + T^{2} ) $
	47	$C_2$	$ ( 1 + T + T^{2} )^{2} $
	53	$C_2$	$ ( 1 - T + T^{2} )^{2} $
	59	$C_2^2$	$ 1 + T^{4} $
	61	$C_2^2$	$ 1 + T^{4} $
	67	$C_2$	$ ( 1 + T^{2} )^{2} $
	71	$C_2$	$ ( 1 - T + T^{2} )^{2} $
	73	$C_2^2$	$ 1 - T^{2} + T^{4} $
	79	$C_2^2$	$ 1 + T^{4} $
	83	$C_2$	$ ( 1 - T + T^{2} )^{2} $
	89	$C_2^2$	$ 1 + T^{4} $
	97	$C_2^2$	$ 1 + T^{4} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−9.584884824197490572710310952119, −9.213562927185144590442048276750, −8.922880485603564792970134179780, −8.188994542937906160475285769996, −7.70235498441598611791804373852, −7.16949797029514871679391394019, −7.04252880988148823826537993481, −6.54996800064447569771127628759, −6.40017030399039791085852546921, −5.94200494687919938743555753875, −5.36385579259932483480011486118, −5.06987677481249570320039762999, −4.85127538154988837202089334076, −4.10494113285929801079505401105, −3.52166644331070202787119388188, −3.02394423742127945405583830380, −2.75089738665443547758264968281, −1.93536883827544989141878694947, −1.92922743326551610108263596566, −0.78077145052131253863577370504, 0.78077145052131253863577370504, 1.92922743326551610108263596566, 1.93536883827544989141878694947, 2.75089738665443547758264968281, 3.02394423742127945405583830380, 3.52166644331070202787119388188, 4.10494113285929801079505401105, 4.85127538154988837202089334076, 5.06987677481249570320039762999, 5.36385579259932483480011486118, 5.94200494687919938743555753875, 6.40017030399039791085852546921, 6.54996800064447569771127628759, 7.04252880988148823826537993481, 7.16949797029514871679391394019, 7.70235498441598611791804373852, 8.188994542937906160475285769996, 8.922880485603564792970134179780, 9.213562927185144590442048276750, 9.584884824197490572710310952119

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(1.387404483\)
\(L(\frac12)\)	\(\approx\)	\(1.387404483\)
\(L(1)\)		not available
\(L(1)\)		not available

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