LMFDB - L-function 2-2240-56.27-c1-0-57

Dirichlet series

L(s) = 1

− 2.84i·3^-s − 5^-s + (0.712 − 2.54i)7^-s − 5.08·9^-s + 5.41·11^-s + 0.641·13^-s + 2.84i·15^-s + 5.16i·17^-s − 3.44i·19^-s + (−7.24 − 2.02i)21^-s − 7.94i·23^-s + 25^-s + 5.94i·27^-s − 10.0i·29^-s − 5.38·31^-s + ⋯

L(s) = 1

− 1.64i·3^-s − 0.447·5^-s + (0.269 − 0.963i)7^-s − 1.69·9^-s + 1.63·11^-s + 0.177·13^-s + 0.734i·15^-s + 1.25i·17^-s − 0.790i·19^-s + (−1.58 − 0.442i)21^-s − 1.65i·23^-s + 0.200·25^-s + 1.14i·27^-s − 1.86i·29^-s − 0.966·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0108i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$-0.999 - 0.0108i$
Analytic conductor:	$17.8864$
Root analytic conductor:	$4.22924$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2240} (671, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2240,\ (\ :1/2),\ -0.999 - 0.0108i)$

Particular Values

$L(1)$	$\approx$	$1.525281170$
$L(\frac12)$	$\approx$	$1.525281170$
$L(\frac{3}{2})$		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 + T $
	7	$ 1 + (-0.712 + 2.54i)T $
good	3	$ 1 + 2.84iT - 3T^{2} $
	11	$ 1 - 5.41T + 11T^{2} $
	13	$ 1 - 0.641T + 13T^{2} $
	17	$ 1 - 5.16iT - 17T^{2} $
	19	$ 1 + 3.44iT - 19T^{2} $
	23	$ 1 + 7.94iT - 23T^{2} $
	29	$ 1 + 10.0iT - 29T^{2} $
	31	$ 1 + 5.38T + 31T^{2} $
	37	$ 1 - 0.931iT - 37T^{2} $
	41	$ 1 + 8.33iT - 41T^{2} $
	43	$ 1 - 6.76T + 43T^{2} $
	47	$ 1 + 8.86T + 47T^{2} $
	53	$ 1 - 6.31iT - 53T^{2} $
	59	$ 1 - 11.5iT - 59T^{2} $
	61	$ 1 - 0.834T + 61T^{2} $
	67	$ 1 - 2.01T + 67T^{2} $
	71	$ 1 + 0.628iT - 71T^{2} $
	73	$ 1 - 10.3iT - 73T^{2} $
	79	$ 1 - 2.98iT - 79T^{2} $
	83	$ 1 - 11.2iT - 83T^{2} $
	89	$ 1 + 13.3iT - 89T^{2} $
	97	$ 1 - 9.07iT - 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

−8.438614717432153759154971631773, −7.78606422756694911848472644005, −7.02961943108793120943655490857, −6.55217105466456632010249158647, −5.88899090353335982665772040951, −4.35844891485890022254857871229, −3.83386852171001370315534848307, −2.42130479814740153059962697668, −1.41347100455587452478380503584, −0.56651625519778270879108286759, 1.62181703729790310379044475976, 3.28712454106691901861784853117, 3.57365257271269988089631425570, 4.67519926552533284568398357453, 5.23189766898437158546936742444, 6.06578803427771479234606013316, 7.10249609383098884671397004771, 8.135779145309390261342274807558, 9.052374262128141341360779293712, 9.293209303826297527870183304994

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.525281170\)
\(L(\frac12)\)	\(\approx\)	\(1.525281170\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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