LMFDB - L-function 2-2240-28.27-c1-0-3

Dirichlet series

L(s) = 1

− 1.73·3^-s + i·5^-s + (1.73 − 2i)7^-s + 3.73i·11^-s − 6.46i·13^-s − 1.73i·15^-s + 0.464i·17^-s − 6·19^-s + (−2.99 + 3.46i)21^-s + 5.46i·23^-s − 25^-s + 5.19·27^-s + 5.92·29^-s − 6·31^-s − 6.46i·33^-s + ⋯

L(s) = 1

− 1.00·3^-s + 0.447i·5^-s + (0.654 − 0.755i)7^-s + 1.12i·11^-s − 1.79i·13^-s − 0.447i·15^-s + 0.112i·17^-s − 1.37·19^-s + (−0.654 + 0.755i)21^-s + 1.13i·23^-s − 0.200·25^-s + 1.00·27^-s + 1.10·29^-s − 1.07·31^-s − 1.12i·33^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$-0.755 - 0.654i$
Analytic conductor:	$17.8864$
Root analytic conductor:	$4.22924$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2240} (1791, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2240,\ (\ :1/2),\ -0.755 - 0.654i)$

Particular Values

$L(1)$	$\approx$	$0.3685791289$
$L(\frac12)$	$\approx$	$0.3685791289$
$L(\frac{3}{2})$		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 - iT $
	7	$ 1 + (-1.73 + 2i)T $
good	3	$ 1 + 1.73T + 3T^{2} $
	11	$ 1 - 3.73iT - 11T^{2} $
	13	$ 1 + 6.46iT - 13T^{2} $
	17	$ 1 - 0.464iT - 17T^{2} $
	19	$ 1 + 6T + 19T^{2} $
	23	$ 1 - 5.46iT - 23T^{2} $
	29	$ 1 - 5.92T + 29T^{2} $
	31	$ 1 + 6T + 31T^{2} $
	37	$ 1 - 2.53T + 37T^{2} $
	41	$ 1 - 3.46iT - 41T^{2} $
	43	$ 1 - 2iT - 43T^{2} $
	47	$ 1 + 1.73T + 47T^{2} $
	53	$ 1 + 2T + 53T^{2} $
	59	$ 1 + 3.46T + 59T^{2} $
	61	$ 1 + 2.53iT - 61T^{2} $
	67	$ 1 - 3.46iT - 67T^{2} $
	71	$ 1 + 0.535iT - 71T^{2} $
	73	$ 1 + 0.928iT - 73T^{2} $
	79	$ 1 - 2.66iT - 79T^{2} $
	83	$ 1 + 8.53T + 83T^{2} $
	89	$ 1 - 9.46iT - 89T^{2} $
	97	$ 1 - 7.39iT - 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

−9.558196989192345444743719136962, −8.299347985411437980330598795799, −7.76314507000293703751203923726, −6.95397503220518694671502161872, −6.19632323610922851707554678945, −5.34292051295322053213082154634, −4.71838031222893140900605262488, −3.73124216282188634410428064241, −2.56037198850972309905948826480, −1.23549733927614951666535601286, 0.15518732293845840692107735632, 1.61391233479248321389818645081, 2.67357958875964586914011678103, 4.18478198315412092916500842794, 4.75636472115034884278332276984, 5.66058028228676386185515788969, 6.25233915471763944964639059085, 6.90434183720621310289871998565, 8.293788988184397033245474664579, 8.662472561421651133782709742641

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.3685791289\)
\(L(\frac12)\)	\(\approx\)	\(0.3685791289\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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