LMFDB - L-function 2-33e2-33.32-c1-0-1

Dirichlet series

L(s) = 1

− 2.43·2^-s + 3.93·4^-s + 3.79i·5^-s + 0.367i·7^-s − 4.71·8^-s − 9.24i·10^-s − 0.948i·13^-s − 0.896i·14^-s + 3.61·16^-s − 3.43·17^-s + 4.26i·19^-s + 14.9i·20^-s + 4.96i·23^-s − 9.40·25^-s + 2.31i·26^-s + ⋯

L(s) = 1

− 1.72·2^-s + 1.96·4^-s + 1.69i·5^-s + 0.139i·7^-s − 1.66·8^-s − 2.92i·10^-s − 0.263i·13^-s − 0.239i·14^-s + 0.904·16^-s − 0.833·17^-s + 0.977i·19^-s + 3.34i·20^-s + 1.03i·23^-s − 1.88·25^-s + 0.453i·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1089$ = $3^{2} \cdot 11^{2}$
Sign:	$-0.972 + 0.231i$
Analytic conductor:	$8.69570$
Root analytic conductor:	$2.94884$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1089} (1088, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1089,\ (\ :1/2),\ -0.972 + 0.231i)$

Particular Values

$L(1)$	$\approx$	$0.2902085954$
$L(\frac12)$	$\approx$	$0.2902085954$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 $
	11	$ 1 $
good	2	$ 1 + 2.43T + 2T^{2} $
	5	$ 1 - 3.79iT - 5T^{2} $
	7	$ 1 - 0.367iT - 7T^{2} $
	13	$ 1 + 0.948iT - 13T^{2} $
	17	$ 1 + 3.43T + 17T^{2} $
	19	$ 1 - 4.26iT - 19T^{2} $
	23	$ 1 - 4.96iT - 23T^{2} $
	29	$ 1 + 2.48T + 29T^{2} $
	31	$ 1 - 3.51T + 31T^{2} $
	37	$ 1 + 7.18T + 37T^{2} $
	41	$ 1 + 2.71T + 41T^{2} $
	43	$ 1 + 1.88iT - 43T^{2} $
	47	$ 1 - 0.0206iT - 47T^{2} $
	53	$ 1 + 5.54iT - 53T^{2} $
	59	$ 1 + 6.62iT - 59T^{2} $
	61	$ 1 - 9.75iT - 61T^{2} $
	67	$ 1 - 4.46T + 67T^{2} $
	71	$ 1 + 10.3iT - 71T^{2} $
	73	$ 1 - 4.39iT - 73T^{2} $
	79	$ 1 - 10.9iT - 79T^{2} $
	83	$ 1 + 9.18T + 83T^{2} $
	89	$ 1 + 3.04iT - 89T^{2} $
	97	$ 1 + 15.0T + 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

−10.19116671087564725155100742111, −9.659825728375226606288910237872, −8.660251266215841459501006329769, −7.88087598729276182783857001913, −7.12191493927450181128236273420, −6.60631919215030036856128197388, −5.63560395915246864087765718958, −3.74460925830925112486475350559, −2.71084964315007959743580427594, −1.76134495542290762331392165882, 0.23600968729111307105706260439, 1.30492980399026857215308234693, 2.41893417288374219102995254190, 4.24956708819211652331820602001, 5.10226307252524744673554936567, 6.36938163211037379726028965745, 7.20852008654909486287121125540, 8.157391774747920734857841004557, 8.816613948352231354422520921037, 9.098465762272512833095743770795

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.2902085954\)
\(L(\frac12)\)	\(\approx\)	\(0.2902085954\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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