LMFDB - L-function 2-5472-76.75-c1-0-41

Dirichlet series

L(s) = 1

+ 0.430·5^-s + 0.0646i·7^-s − 2.19i·11^-s − 3.40i·13^-s − 3.67·17^-s + (−1.70 + 4.00i)19^-s + 2.69i·23^-s − 4.81·25^-s + 3.11i·29^-s + 7.11·31^-s + 0.0278i·35^-s + 8.91i·37^-s + 7.22i·41^-s + 3.12i·43^-s − 10.6i·47^-s + ⋯

L(s) = 1

+ 0.192·5^-s + 0.0244i·7^-s − 0.660i·11^-s − 0.943i·13^-s − 0.890·17^-s + (−0.392 + 0.919i)19^-s + 0.561i·23^-s − 0.962·25^-s + 0.578i·29^-s + 1.27·31^-s + 0.00470i·35^-s + 1.46i·37^-s + 1.12i·41^-s + 0.475i·43^-s − 1.55i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5472$ = $2^{5} \cdot 3^{2} \cdot 19$
Sign:	$0.927 - 0.373i$
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:	$(2,\ 5472,\ (\ :1/2),\ 0.927 - 0.373i)$

Particular Values

$L(1)$	$\approx$	$1.745604778$
$L(\frac12)$	$\approx$	$1.745604778$
$L(\frac{3}{2})$		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 + (1.70 - 4.00i)T $
good	5	$ 1 - 0.430T + 5T^{2} $
	7	$ 1 - 0.0646iT - 7T^{2} $
	11	$ 1 + 2.19iT - 11T^{2} $
	13	$ 1 + 3.40iT - 13T^{2} $
	17	$ 1 + 3.67T + 17T^{2} $
	23	$ 1 - 2.69iT - 23T^{2} $
	29	$ 1 - 3.11iT - 29T^{2} $
	31	$ 1 - 7.11T + 31T^{2} $
	37	$ 1 - 8.91iT - 37T^{2} $
	41	$ 1 - 7.22iT - 41T^{2} $
	43	$ 1 - 3.12iT - 43T^{2} $
	47	$ 1 + 10.6iT - 47T^{2} $
	53	$ 1 - 7.22iT - 53T^{2} $
	59	$ 1 - 8.33T + 59T^{2} $
	61	$ 1 - 10.8T + 61T^{2} $
	67	$ 1 - 6.16T + 67T^{2} $
	71	$ 1 + 9.12T + 71T^{2} $
	73	$ 1 - 14.6T + 73T^{2} $
	79	$ 1 + 0.185T + 79T^{2} $
	83	$ 1 + 13.6iT - 83T^{2} $
	89	$ 1 + 7.58iT - 89T^{2} $
	97	$ 1 + 0.175iT - 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

−8.353734885712481945474189383612, −7.58340712844542060754238903231, −6.67324748720480450645538907454, −6.04103862141652673946903164677, −5.42039764050385015134900431878, −4.55780195883235444444624945565, −3.66977762557929022054350115042, −2.92323972740743389609756781550, −1.94960873758521603294415980908, −0.816997921289788041832982548974, 0.59693980802212499007201093319, 2.13679320271752303178111415263, 2.38781928579597058239168774775, 3.90415518837945410981271987971, 4.33542953781520724128641662820, 5.16258184050922141005969768168, 6.06352303621471804534420260262, 6.79454312040497086797929494662, 7.20740569229008325177020767227, 8.216267644952524518649048762836

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.745604778\)
\(L(\frac12)\)	\(\approx\)	\(1.745604778\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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