LMFDB - L-function 2-5472-76.75-c1-0-58

Dirichlet series

L(s) = 1

− 4.07·5^-s − 1.30i·7^-s − 4.01i·11^-s + 4.82i·13^-s + 2.18·17^-s + (−3.91 + 1.91i)19^-s − 3.65i·23^-s + 11.5·25^-s + 9.24i·29^-s + 4.74·31^-s + 5.29i·35^-s + 0.423i·37^-s − 3.88i·41^-s + 4.97i·43^-s − 4.57i·47^-s + ⋯

L(s) = 1

− 1.82·5^-s − 0.491i·7^-s − 1.21i·11^-s + 1.33i·13^-s + 0.530·17^-s + (−0.898 + 0.438i)19^-s − 0.762i·23^-s + 2.31·25^-s + 1.71i·29^-s + 0.852·31^-s + 0.895i·35^-s + 0.0696i·37^-s − 0.606i·41^-s + 0.758i·43^-s − 0.667i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5472$ = $2^{5} \cdot 3^{2} \cdot 19$
Sign:	$-0.325 + 0.945i$
Analytic conductor:	$43.6941$
Root analytic conductor:	$6.61015$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5472} (2431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5472,\ (\ :1/2),\ -0.325 + 0.945i)$

Particular Values

$L(1)$	$\approx$	$0.6627388816$
$L(\frac12)$	$\approx$	$0.6627388816$
$L(\frac{3}{2})$		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 + (3.91 - 1.91i)T $
good	5	$ 1 + 4.07T + 5T^{2} $
	7	$ 1 + 1.30iT - 7T^{2} $
	11	$ 1 + 4.01iT - 11T^{2} $
	13	$ 1 - 4.82iT - 13T^{2} $
	17	$ 1 - 2.18T + 17T^{2} $
	23	$ 1 + 3.65iT - 23T^{2} $
	29	$ 1 - 9.24iT - 29T^{2} $
	31	$ 1 - 4.74T + 31T^{2} $
	37	$ 1 - 0.423iT - 37T^{2} $
	41	$ 1 + 3.88iT - 41T^{2} $
	43	$ 1 - 4.97iT - 43T^{2} $
	47	$ 1 + 4.57iT - 47T^{2} $
	53	$ 1 - 1.30iT - 53T^{2} $
	59	$ 1 - 10.7T + 59T^{2} $
	61	$ 1 - 8.76T + 61T^{2} $
	67	$ 1 + 8.77T + 67T^{2} $
	71	$ 1 + 14.2T + 71T^{2} $
	73	$ 1 + 5.08T + 73T^{2} $
	79	$ 1 + 13.6T + 79T^{2} $
	83	$ 1 + 12.3iT - 83T^{2} $
	89	$ 1 - 2.16iT - 89T^{2} $
	97	$ 1 - 12.4iT - 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

−8.002889411502564862185658251251, −7.18298285796748299069456855955, −6.78116320430459371925545770589, −5.86130516035310092192220587489, −4.75291676685955014594108264666, −4.14933983091594888009305227036, −3.61209696335105225515939535601, −2.80690586784032729239504339719, −1.30899626518295274481406873297, −0.24913078586270595832230198631, 0.850337602394558813770792442801, 2.36462373169069023467704089765, 3.13754344646280305685210467929, 4.05261275131047225754328599958, 4.53774552901656273807343592984, 5.40676840552402766367758237261, 6.26559112070622395197406512782, 7.37368167029903263281239774514, 7.50251034020693214648060823660, 8.343241457021477786900769112914

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6627388816\)
\(L(\frac12)\)	\(\approx\)	\(0.6627388816\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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