LMFDB - L-function 2-72e2-8.5-c1-0-57

Dirichlet series

L(s) = 1

− 3.87i·5^-s + 3.87·7^-s + 3i·11^-s + 3.87i·13^-s − 2i·19^-s + 3.87·23^-s − 10.0·25^-s − 3.87i·29^-s + 3.87·31^-s − 15.0i·35^-s + 7.74i·37^-s + 9·41^-s − 7i·43^-s + 3.87·47^-s + 8.00·49^-s + ⋯

L(s) = 1

− 1.73i·5^-s + 1.46·7^-s + 0.904i·11^-s + 1.07i·13^-s − 0.458i·19^-s + 0.807·23^-s − 2.00·25^-s − 0.719i·29^-s + 0.695·31^-s − 2.53i·35^-s + 1.27i·37^-s + 1.40·41^-s − 1.06i·43^-s + 0.564·47^-s + 1.14·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5184$ = $2^{6} \cdot 3^{4}$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$41.3944$
Root analytic conductor:	$6.43385$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5184} (2593, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5184,\ (\ :1/2),\ 0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$2.511041775$
$L(\frac12)$	$\approx$	$2.511041775$
$L(\frac{3}{2})$		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 $
good	5	$ 1 + 3.87iT - 5T^{2} $
	7	$ 1 - 3.87T + 7T^{2} $
	11	$ 1 - 3iT - 11T^{2} $
	13	$ 1 - 3.87iT - 13T^{2} $
	17	$ 1 + 17T^{2} $
	19	$ 1 + 2iT - 19T^{2} $
	23	$ 1 - 3.87T + 23T^{2} $
	29	$ 1 + 3.87iT - 29T^{2} $
	31	$ 1 - 3.87T + 31T^{2} $
	37	$ 1 - 7.74iT - 37T^{2} $
	41	$ 1 - 9T + 41T^{2} $
	43	$ 1 + 7iT - 43T^{2} $
	47	$ 1 - 3.87T + 47T^{2} $
	53	$ 1 + 7.74iT - 53T^{2} $
	59	$ 1 + 3iT - 59T^{2} $
	61	$ 1 - 11.6iT - 61T^{2} $
	67	$ 1 - 11iT - 67T^{2} $
	71	$ 1 - 7.74T + 71T^{2} $
	73	$ 1 - 8T + 73T^{2} $
	79	$ 1 + 3.87T + 79T^{2} $
	83	$ 1 + 3iT - 83T^{2} $
	89	$ 1 + 12T + 89T^{2} $
	97	$ 1 - T + 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

−8.317767553111854021297556134936, −7.52558440187243330493730740563, −6.82379621521928666909184859220, −5.68793326382312851355371499749, −4.99877125473703777248970250041, −4.52033187249174691141340100053, −4.09897782843367829277407432733, −2.39537292370687170905673875542, −1.62959433602507900189961485757, −0.874259496250278819102729167889, 0.954766117641276624792135858477, 2.19682465392589172178157380561, 2.96487516212965279805701137204, 3.61222971903952200281175538646, 4.64211820488592452604053629191, 5.55929686803401364751719266920, 6.08658120399765735013318597989, 6.94730353579087914398651472784, 7.74935395234492751038821104243, 7.970859110294578441933457794290

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(2.511041775\)
\(L(\frac12)\)	\(\approx\)	\(2.511041775\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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