LMFDB - L-function 2-2240-40.29-c1-0-35

Dirichlet series

L(s) = 1

− 1.03·3^-s + (−0.188 − 2.22i)5^-s + i·7^-s − 1.92·9^-s + 3.68i·11^-s + 5.30·13^-s + (0.195 + 2.30i)15^-s − 5.49i·17^-s − 1.62i·19^-s − 1.03i·21^-s + 1.24i·23^-s + (−4.92 + 0.839i)25^-s + 5.10·27^-s + 7.17i·29^-s − 2.06·31^-s + ⋯

L(s) = 1

− 0.597·3^-s + (−0.0842 − 0.996i)5^-s + 0.377i·7^-s − 0.642·9^-s + 1.11i·11^-s + 1.47·13^-s + (0.0503 + 0.595i)15^-s − 1.33i·17^-s − 0.372i·19^-s − 0.225i·21^-s + 0.259i·23^-s + (−0.985 + 0.167i)25^-s + 0.981·27^-s + 1.33i·29^-s − 0.371·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$0.764 + 0.644i$
Analytic conductor:	$17.8864$
Root analytic conductor:	$4.22924$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2240} (1569, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2240,\ (\ :1/2),\ 0.764 + 0.644i)$

Particular Values

$L(1)$	$\approx$	$1.261131239$
$L(\frac12)$	$\approx$	$1.261131239$
$L(\frac{3}{2})$		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 + (0.188 + 2.22i)T $
	7	$ 1 - iT $
good	3	$ 1 + 1.03T + 3T^{2} $
	11	$ 1 - 3.68iT - 11T^{2} $
	13	$ 1 - 5.30T + 13T^{2} $
	17	$ 1 + 5.49iT - 17T^{2} $
	19	$ 1 + 1.62iT - 19T^{2} $
	23	$ 1 - 1.24iT - 23T^{2} $
	29	$ 1 - 7.17iT - 29T^{2} $
	31	$ 1 + 2.06T + 31T^{2} $
	37	$ 1 + 5.85T + 37T^{2} $
	41	$ 1 - 2T + 41T^{2} $
	43	$ 1 - 10.5T + 43T^{2} $
	47	$ 1 + 0.928iT - 47T^{2} $
	53	$ 1 - 6.61T + 53T^{2} $
	59	$ 1 - 4.23iT - 59T^{2} $
	61	$ 1 + 8.59iT - 61T^{2} $
	67	$ 1 - 9.30T + 67T^{2} $
	71	$ 1 + 3.74T + 71T^{2} $
	73	$ 1 + 10.5iT - 73T^{2} $
	79	$ 1 - 9.63T + 79T^{2} $
	83	$ 1 - 4.84T + 83T^{2} $
	89	$ 1 - 12.6T + 89T^{2} $
	97	$ 1 + 12.3iT - 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

−9.063694999334214820827474070032, −8.321957958199755751868014175464, −7.37051551624866360001496676393, −6.55258404029816393778519778858, −5.55892888089673053236732610143, −5.15497367853428286705189370994, −4.27093651241178850303631068810, −3.15738465873212702769645647099, −1.86949305910042269963600738626, −0.66345743022687287194162179831, 0.869425635117852436188195720937, 2.37447515356949653604174520290, 3.56100716565266708686533582513, 3.94186374826330286035033611444, 5.49170602468842013910259293804, 6.16319660358543787727444799812, 6.40224742283588565328241841418, 7.63180955610219513249793017254, 8.334533304367400805443613032453, 8.949484287254005881149645581902

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.261131239\)
\(L(\frac12)\)	\(\approx\)	\(1.261131239\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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