LMFDB - L-function 2-33e2-11.10-c2-0-22

Dirichlet series

L(s) = 1

− 3.86i·2^-s − 10.9·4^-s − 4.15·5^-s − 7.44i·7^-s + 26.7i·8^-s + 16.0i·10^-s + 7.32i·13^-s − 28.7·14^-s + 59.6·16^-s + 3.57i·17^-s + 29.9i·19^-s + 45.3·20^-s + 20.6·23^-s − 7.77·25^-s + 28.2·26^-s + ⋯

L(s) = 1

− 1.93i·2^-s − 2.73·4^-s − 0.830·5^-s − 1.06i·7^-s + 3.34i·8^-s + 1.60i·10^-s + 0.563i·13^-s − 2.05·14^-s + 3.72·16^-s + 0.210i·17^-s + 1.57i·19^-s + 2.26·20^-s + 0.897·23^-s − 0.310·25^-s + 1.08·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1089$ = $3^{2} \cdot 11^{2}$
Sign:	$-0.372 + 0.927i$
Analytic conductor:	$29.6731$
Root analytic conductor:	$5.44730$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1089} (604, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1089,\ (\ :1),\ -0.372 + 0.927i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.9941370652$
$L(\frac12)$	$\approx$	$0.9941370652$
$L(2)$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 $
	11	$ 1 $
good	2	$ 1 + 3.86iT - 4T^{2} $
	5	$ 1 + 4.15T + 25T^{2} $
	7	$ 1 + 7.44iT - 49T^{2} $
	13	$ 1 - 7.32iT - 169T^{2} $
	17	$ 1 - 3.57iT - 289T^{2} $
	19	$ 1 - 29.9iT - 361T^{2} $
	23	$ 1 - 20.6T + 529T^{2} $
	29	$ 1 + 17.1iT - 841T^{2} $
	31	$ 1 - 15.7T + 961T^{2} $
	37	$ 1 + 32.2T + 1.36e3T^{2} $
	41	$ 1 + 16.2iT - 1.68e3T^{2} $
	43	$ 1 - 33.7iT - 1.84e3T^{2} $
	47	$ 1 - 38.9T + 2.20e3T^{2} $
	53	$ 1 - 36.6T + 2.80e3T^{2} $
	59	$ 1 - 69.9T + 3.48e3T^{2} $
	61	$ 1 + 22.7iT - 3.72e3T^{2} $
	67	$ 1 + 63.0T + 4.48e3T^{2} $
	71	$ 1 + 31.5T + 5.04e3T^{2} $
	73	$ 1 - 103. iT - 5.32e3T^{2} $
	79	$ 1 - 18.2iT - 6.24e3T^{2} $
	83	$ 1 + 118. iT - 6.88e3T^{2} $
	89	$ 1 + 154.T + 7.92e3T^{2} $
	97	$ 1 + 69.2T + 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

−9.862850100803278184655477532167, −8.789612311560181945122692795173, −8.092276021416552144293016693780, −7.18487854225195594766295110108, −5.63787244504762045176835068079, −4.35534646507644500140743975216, −4.00419387992187502317995405577, −3.14135803659508131320746305090, −1.79700337375762258415070618505, −0.74236240148438108040507305194, 0.50916670115686511173168558726, 2.98397170614802090944027441430, 4.19761311270855394813731389499, 5.11561844857868793404082670749, 5.66093095491377870963929198312, 6.78656601714160551943839633055, 7.28568409538325005055802378972, 8.213700364867354692585487978807, 8.811890794924628031461547635373, 9.337947416399371879083216955908

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.9941370652\)
\(L(\frac12)\)	\(\approx\)	\(0.9941370652\)
\(L(2)\)		not available
\(L(1)\)		not available

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