LMFDB - L-function 2-1152-12.11-c1-0-10

Dirichlet series

L(s) = 1

− 0.585i·5^-s − 0.828i·7^-s + 2.82·11^-s − 2.82·13^-s + 2.58i·17^-s − 5.65i·19^-s + 6.82·23^-s + 4.65·25^-s − 3.41i·29^-s − 8.82i·31^-s − 0.485·35^-s − 7.65·37^-s + 5.41i·41^-s − 1.65i·43^-s − 4.48·47^-s + ⋯

L(s) = 1

− 0.261i·5^-s − 0.313i·7^-s + 0.852·11^-s − 0.784·13^-s + 0.627i·17^-s − 1.29i·19^-s + 1.42·23^-s + 0.931·25^-s − 0.634i·29^-s − 1.58i·31^-s − 0.0820·35^-s − 1.25·37^-s + 0.845i·41^-s − 0.252i·43^-s − 0.654·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1152$ = $2^{7} \cdot 3^{2}$
Sign:	$0.577 + 0.816i$
Analytic conductor:	$9.19876$
Root analytic conductor:	$3.03294$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1152} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1152,\ (\ :1/2),\ 0.577 + 0.816i)$

Particular Values

$L(1)$	$\approx$	$1.586923297$
$L(\frac12)$	$\approx$	$1.586923297$
$L(\frac{3}{2})$		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 + 0.585iT - 5T^{2} $
	7	$ 1 + 0.828iT - 7T^{2} $
	11	$ 1 - 2.82T + 11T^{2} $
	13	$ 1 + 2.82T + 13T^{2} $
	17	$ 1 - 2.58iT - 17T^{2} $
	19	$ 1 + 5.65iT - 19T^{2} $
	23	$ 1 - 6.82T + 23T^{2} $
	29	$ 1 + 3.41iT - 29T^{2} $
	31	$ 1 + 8.82iT - 31T^{2} $
	37	$ 1 + 7.65T + 37T^{2} $
	41	$ 1 - 5.41iT - 41T^{2} $
	43	$ 1 + 1.65iT - 43T^{2} $
	47	$ 1 + 4.48T + 47T^{2} $
	53	$ 1 + 9.07iT - 53T^{2} $
	59	$ 1 - 13.6T + 59T^{2} $
	61	$ 1 + 3.65T + 61T^{2} $
	67	$ 1 + 12iT - 67T^{2} $
	71	$ 1 - 12.4T + 71T^{2} $
	73	$ 1 - 4T + 73T^{2} $
	79	$ 1 + 10.4iT - 79T^{2} $
	83	$ 1 + 10.8T + 83T^{2} $
	89	$ 1 - 3.75iT - 89T^{2} $
	97	$ 1 - 2.34T + 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

−9.530116024089088146449229527000, −8.983402835262401286249471434995, −8.071245743023160315368999158165, −7.05022328978353077105683881979, −6.53131637681779322283367766950, −5.25378497952992784352540967552, −4.53532908422761197339466836864, −3.48561489368504187856046285830, −2.25738184363746045780969382042, −0.77942883312104095853371183816, 1.33243344650193482672169034751, 2.72677888271569562030243528530, 3.64912377682849718249442880637, 4.87537088238943367927189521620, 5.58711389278937596687551122010, 6.89905867450471473372043497959, 7.11369266486563964833876105087, 8.495111333392880487740433978438, 9.012058044242434787843157940692, 9.944662754015245299189043331265

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.586923297\)
\(L(\frac12)\)	\(\approx\)	\(1.586923297\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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