LMFDB - L-function 2-2240-5.4-c1-0-56

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0685 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0685 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$-0.0685 + 0.997i$
Analytic conductor:	$17.8864$
Root analytic conductor:	$4.22924$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2240} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2240,\ (\ :1/2),\ -0.0685 + 0.997i)$

Particular Values

$L(1)$	$\approx$	$2.236251619$
$L(\frac12)$	$\approx$	$2.236251619$
$L(\frac{3}{2})$		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

−8.642085437076022814712184732416, −8.020888037245485233847729176658, −7.14967478338034373372057468255, −6.68434926449736713470216098808, −5.62113521483438804999809782457, −5.20958585618300697209212661120, −3.78505952473568518584081039736, −2.76092098946524273936329291759, −1.77266963744978042878225665795, −0.829180366314233630554997712580, 1.40733213691693121580229138562, 2.51946704953069280265439512749, 3.50794331323079545501102134832, 4.56663580702425188289339607447, 5.22658152000734640705237471988, 5.86133917107501464864974112614, 6.94892576500806800972157773663, 7.57967819154826894836412378166, 8.951630052369437707927191645875, 9.310796612885238195656222895864

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.236251619\)
\(L(\frac12)\)	\(\approx\)	\(2.236251619\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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