LMFDB - L-function 2-1134-21.20-c1-0-24

Dirichlet series

L(s) = 1

− i·2^-s − 4^-s − 1.01·5^-s + (0.141 + 2.64i)7^-s + i·8^-s + 1.01i·10^-s + 5.07i·11^-s − 5.45i·13^-s + (2.64 − 0.141i)14^-s + 16^-s − 6.19·17^-s − 7.96i·19^-s + 1.01·20^-s + 5.07·22^-s − 0.835i·23^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s − 0.456·5^-s + (0.0533 + 0.998i)7^-s + 0.353i·8^-s + 0.322i·10^-s + 1.53i·11^-s − 1.51i·13^-s + (0.706 − 0.0377i)14^-s + 0.250·16^-s − 1.50·17^-s − 1.82i·19^-s + 0.228·20^-s + 1.08·22^-s − 0.174i·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0533i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1134$ = $2 \cdot 3^{4} \cdot 7$
Sign:	$-0.998 + 0.0533i$
Analytic conductor:	$9.05503$
Root analytic conductor:	$3.00915$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1134} (1133, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1134,\ (\ :1/2),\ -0.998 + 0.0533i)$

Particular Values

$L(1)$	$\approx$	$0.4169806233$
$L(\frac12)$	$\approx$	$0.4169806233$
$L(\frac{3}{2})$		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 + iT $
	3	$ 1 $
	7	$ 1 + (-0.141 - 2.64i)T $
good	5	$ 1 + 1.01T + 5T^{2} $
	11	$ 1 - 5.07iT - 11T^{2} $
	13	$ 1 + 5.45iT - 13T^{2} $
	17	$ 1 + 6.19T + 17T^{2} $
	19	$ 1 + 7.96iT - 19T^{2} $
	23	$ 1 + 0.835iT - 23T^{2} $
	29	$ 1 + 8.91iT - 29T^{2} $
	31	$ 1 + 3.97iT - 31T^{2} $
	37	$ 1 + 2.64T + 37T^{2} $
	41	$ 1 + 6.36T + 41T^{2} $
	43	$ 1 - 5.67T + 43T^{2} $
	47	$ 1 - 0.563T + 47T^{2} $
	53	$ 1 - 2.19iT - 53T^{2} $
	59	$ 1 + 7.21T + 59T^{2} $
	61	$ 1 - 7.94iT - 61T^{2} $
	67	$ 1 - 10.7T + 67T^{2} $
	71	$ 1 - 6.20iT - 71T^{2} $
	73	$ 1 + 7.05iT - 73T^{2} $
	79	$ 1 + 4.74T + 79T^{2} $
	83	$ 1 + 11.0T + 83T^{2} $
	89	$ 1 + 2.11T + 89T^{2} $
	97	$ 1 - 2.78iT - 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

−9.489666470191019876539486761067, −8.739427521607537186250459233602, −7.907376566323412669002759517599, −7.02185877273648368008756911960, −5.87472995131015628243212248708, −4.85514206187504859933073319912, −4.20783121390404435069828380658, −2.75939629002308664853311470378, −2.15461116268021019294450992368, −0.17954914318116927941039594637, 1.55445873505594662591945532676, 3.53039117851586972743464016356, 4.04833389490433320476037165456, 5.11365094992438173623822136010, 6.27314198179820861703573584829, 6.81927776580515958067358226716, 7.72928386574928860188700882772, 8.523122809112734309243876351818, 9.114337902786441824751116229354, 10.22836175878567294033152861364

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.4169806233\)
\(L(\frac12)\)	\(\approx\)	\(0.4169806233\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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