LMFDB - L-function 2-1152-8.3-c2-0-11

Dirichlet series

L(s) = 1

− 0.898i·5^-s − 2.82i·7^-s + 4.38·11^-s + 13.7i·13^-s − 17.5·17^-s + 4.38·19^-s + 22.0i·23^-s + 24.1·25^-s − 44.4i·29^-s + 53.1i·31^-s − 2.54·35^-s + 35.1i·37^-s − 37.5·41^-s + 49.6·43^-s − 38.4i·47^-s + ⋯

L(s) = 1

− 0.179i·5^-s − 0.404i·7^-s + 0.398·11^-s + 1.06i·13^-s − 1.03·17^-s + 0.230·19^-s + 0.958i·23^-s + 0.967·25^-s − 1.53i·29^-s + 1.71i·31^-s − 0.0726·35^-s + 0.951i·37^-s − 0.916·41^-s + 1.15·43^-s − 0.818i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1152$ = $2^{7} \cdot 3^{2}$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$31.3897$
Root analytic conductor:	$5.60265$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1152} (703, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1152,\ (\ :1),\ 0.707 - 0.707i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.731952887$
$L(\frac12)$	$\approx$	$1.731952887$
$L(2)$		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 + 0.898iT - 25T^{2} $
	7	$ 1 + 2.82iT - 49T^{2} $
	11	$ 1 - 4.38T + 121T^{2} $
	13	$ 1 - 13.7iT - 169T^{2} $
	17	$ 1 + 17.5T + 289T^{2} $
	19	$ 1 - 4.38T + 361T^{2} $
	23	$ 1 - 22.0iT - 529T^{2} $
	29	$ 1 + 44.4iT - 841T^{2} $
	31	$ 1 - 53.1iT - 961T^{2} $
	37	$ 1 - 35.1iT - 1.36e3T^{2} $
	41	$ 1 + 37.5T + 1.68e3T^{2} $
	43	$ 1 - 49.6T + 1.84e3T^{2} $
	47	$ 1 + 38.4iT - 2.20e3T^{2} $
	53	$ 1 - 1.70iT - 2.80e3T^{2} $
	59	$ 1 - 34.6T + 3.48e3T^{2} $
	61	$ 1 + 24.4iT - 3.72e3T^{2} $
	67	$ 1 - 93.7T + 4.48e3T^{2} $
	71	$ 1 - 123. iT - 5.04e3T^{2} $
	73	$ 1 + 10T + 5.32e3T^{2} $
	79	$ 1 - 131. iT - 6.24e3T^{2} $
	83	$ 1 - 110.T + 6.88e3T^{2} $
	89	$ 1 - 73.1T + 7.92e3T^{2} $
	97	$ 1 + 105.T + 9.40e3T^{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

−9.621516694538714251641587528533, −8.929308628575563465412070487697, −8.178487324146350223028117265196, −6.98135282074376464107141633069, −6.63755572626861030052447522940, −5.37010586192481237782519979062, −4.45704837217704206936962063534, −3.64946690769794039457785569738, −2.29078020538055217567660252048, −1.07952227682514234273779746952, 0.60859346707266048397303410690, 2.18569705934990931900120175901, 3.14547347215066988069520483740, 4.28762488847794343008658520324, 5.26550406282137635494512324295, 6.15534029376808959964269055066, 6.97978921706370354005670495175, 7.87952322668835523352987544286, 8.806038160377762976010464929018, 9.327361288408794287107706511247

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.731952887\)
\(L(\frac12)\)	\(\approx\)	\(1.731952887\)
\(L(2)\)		not available
\(L(1)\)		not available

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