LMFDB - L-function 2-2240-40.29-c1-0-28

Dirichlet series

L(s) = 1

− 3.11·3^-s + (−2.08 − 0.809i)5^-s − i·7^-s + 6.68·9^-s − 2.64i·11^-s + 0.479·13^-s + (6.48 + 2.52i)15^-s + 1.49i·17^-s − 2.16i·19^-s + 3.11i·21^-s + 6.33i·23^-s + (3.68 + 3.37i)25^-s − 11.4·27^-s + 5.25i·29^-s − 6.22·31^-s + ⋯

L(s) = 1

− 1.79·3^-s + (−0.932 − 0.362i)5^-s − 0.377i·7^-s + 2.22·9^-s − 0.798i·11^-s + 0.133·13^-s + (1.67 + 0.650i)15^-s + 0.362i·17^-s − 0.497i·19^-s + 0.679i·21^-s + 1.32i·23^-s + (0.737 + 0.675i)25^-s − 2.20·27^-s + 0.976i·29^-s − 1.11·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.403 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$0.403 + 0.915i$
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:	$(2,\ 2240,\ (\ :1/2),\ 0.403 + 0.915i)$

Particular Values

$L(1)$	$\approx$	$0.5629839002$
$L(\frac12)$	$\approx$	$0.5629839002$
$L(\frac{3}{2})$		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 + (2.08 + 0.809i)T $
	7	$ 1 + iT $
good	3	$ 1 + 3.11T + 3T^{2} $
	11	$ 1 + 2.64iT - 11T^{2} $
	13	$ 1 - 0.479T + 13T^{2} $
	17	$ 1 - 1.49iT - 17T^{2} $
	19	$ 1 + 2.16iT - 19T^{2} $
	23	$ 1 - 6.33iT - 23T^{2} $
	29	$ 1 - 5.25iT - 29T^{2} $
	31	$ 1 + 6.22T + 31T^{2} $
	37	$ 1 - 11.3T + 37T^{2} $
	41	$ 1 - 2T + 41T^{2} $
	43	$ 1 + 9.98T + 43T^{2} $
	47	$ 1 + 7.68iT - 47T^{2} $
	53	$ 1 + 3.04T + 53T^{2} $
	59	$ 1 - 9.20iT - 59T^{2} $
	61	$ 1 - 10.8iT - 61T^{2} $
	67	$ 1 - 9.73T + 67T^{2} $
	71	$ 1 - 0.525T + 71T^{2} $
	73	$ 1 + 9.98iT - 73T^{2} $
	79	$ 1 - 13.9T + 79T^{2} $
	83	$ 1 - 11.3T + 83T^{2} $
	89	$ 1 - 2.95T + 89T^{2} $
	97	$ 1 + 7.97iT - 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

−8.926510201535119895998699986353, −7.912173688267801421277604551386, −7.22473546448708955203727166014, −6.51178607647547356729350900440, −5.62193024111443833190886207944, −5.06314238640949545051470989325, −4.16858016554290999017631585904, −3.41383530058951359224650160604, −1.39836020346705741124725404928, −0.43929898272620759375999701245, 0.70965581210636465295987588997, 2.23109067361082671130569412224, 3.71526470854603785310872365243, 4.58445817960482672510967438907, 5.11098730546151999393826694993, 6.26074698711723422993715563663, 6.55680651309639295594541134132, 7.53267334804530228746117688300, 8.125783410620382399658232798934, 9.449948025719948784059813801823

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.5629839002\)
\(L(\frac12)\)	\(\approx\)	\(0.5629839002\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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