LMFDB - L-function 2-1120-5.4-c1-0-11

Dirichlet series

L(s) = 1

− 1.63i·3^-s + (−2.23 − 0.153i)5^-s + i·7^-s + 0.328·9^-s − 1.24·11^-s + 4.20i·13^-s + (−0.250 + 3.64i)15^-s + 3.39i·17^-s + 6.46·19^-s + 1.63·21^-s + 2.15i·23^-s + (4.95 + 0.683i)25^-s − 5.44i·27^-s − 3.96·29^-s + 10.0·31^-s + ⋯

L(s) = 1

− 0.943i·3^-s + (−0.997 − 0.0685i)5^-s + 0.377i·7^-s + 0.109·9^-s − 0.374·11^-s + 1.16i·13^-s + (−0.0646 + 0.941i)15^-s + 0.823i·17^-s + 1.48·19^-s + 0.356·21^-s + 0.449i·23^-s + (0.990 + 0.136i)25^-s − 1.04i·27^-s − 0.736·29^-s + 1.80·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0685i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1120$ = $2^{5} \cdot 5 \cdot 7$
Sign:	$0.997 + 0.0685i$
Analytic conductor:	$8.94324$
Root analytic conductor:	$2.99052$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1120} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1120,\ (\ :1/2),\ 0.997 + 0.0685i)$

Particular Values

$L(1)$	$\approx$	$1.333536921$
$L(\frac12)$	$\approx$	$1.333536921$
$L(\frac{3}{2})$		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 $
	5	$ 1 + (2.23 + 0.153i)T $
	7	$ 1 - iT $
good	3	$ 1 + 1.63iT - 3T^{2} $
	11	$ 1 + 1.24T + 11T^{2} $
	13	$ 1 - 4.20iT - 13T^{2} $
	17	$ 1 - 3.39iT - 17T^{2} $
	19	$ 1 - 6.46T + 19T^{2} $
	23	$ 1 - 2.15iT - 23T^{2} $
	29	$ 1 + 3.96T + 29T^{2} $
	31	$ 1 - 10.0T + 31T^{2} $
	37	$ 1 + 6.76iT - 37T^{2} $
	41	$ 1 + 0.131T + 41T^{2} $
	43	$ 1 + 7.40iT - 43T^{2} $
	47	$ 1 - 4.82iT - 47T^{2} $
	53	$ 1 - 10.0iT - 53T^{2} $
	59	$ 1 - 10.9T + 59T^{2} $
	61	$ 1 + 6.33T + 61T^{2} $
	67	$ 1 - 2.65iT - 67T^{2} $
	71	$ 1 + 0.754T + 71T^{2} $
	73	$ 1 - 6.03iT - 73T^{2} $
	79	$ 1 - 14.6T + 79T^{2} $
	83	$ 1 - 14.0iT - 83T^{2} $
	89	$ 1 - 12.6T + 89T^{2} $
	97	$ 1 + 0.914iT - 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

−9.719083030311344402702618136971, −8.861615607491800256126106358911, −7.964311054725694117851497855703, −7.41045137809122897598605450430, −6.69203538277729928321727426175, −5.68375639826399958964507200149, −4.53954529580680138688536212385, −3.61876102259087331397881670833, −2.31768480279854528430991147506, −1.08046439096713055611024755420, 0.76051904040828307288564385906, 2.96693950208420419442696808021, 3.58542445427792752040607136376, 4.72280397031164105487027960232, 5.15879105180176691363170630157, 6.58681468923159314117382433071, 7.58799012780162969530735674772, 8.039554746915273597197866426081, 9.144979207420570175852871518402, 10.05627704733567137109369978014

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.333536921\)
\(L(\frac12)\)	\(\approx\)	\(1.333536921\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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