LMFDB - L-function 2-2240-5.4-c1-0-24

Dirichlet series

L(s) = 1

− 3.12i·3^-s + (−1.32 + 1.80i)5^-s + i·7^-s − 6.76·9^-s + 2.48·11^-s − 4.15i·13^-s + (5.64 + 4.12i)15^-s + 5.76i·17^-s + 1.60·19^-s + 3.12·21^-s + 7.28i·23^-s + (−1.51 − 4.76i)25^-s + 11.7i·27^-s + 1.45·29^-s + 2.24·31^-s + ⋯

L(s) = 1

− 1.80i·3^-s + (−0.590 + 0.807i)5^-s + 0.377i·7^-s − 2.25·9^-s + 0.749·11^-s − 1.15i·13^-s + (1.45 + 1.06i)15^-s + 1.39i·17^-s + 0.369·19^-s + 0.681·21^-s + 1.51i·23^-s + (−0.303 − 0.952i)25^-s + 2.26i·27^-s + 0.270·29^-s + 0.404·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 + 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2240$ = $2^{6} \cdot 5 \cdot 7$
Sign:	$0.807 + 0.590i$
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:	$(2,\ 2240,\ (\ :1/2),\ 0.807 + 0.590i)$

Particular Values

$L(1)$	$\approx$	$1.464000867$
$L(\frac12)$	$\approx$	$1.464000867$
$L(\frac{3}{2})$		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 + (1.32 - 1.80i)T $
	7	$ 1 - iT $
good	3	$ 1 + 3.12iT - 3T^{2} $
	11	$ 1 - 2.48T + 11T^{2} $
	13	$ 1 + 4.15iT - 13T^{2} $
	17	$ 1 - 5.76iT - 17T^{2} $
	19	$ 1 - 1.60T + 19T^{2} $
	23	$ 1 - 7.28iT - 23T^{2} $
	29	$ 1 - 1.45T + 29T^{2} $
	31	$ 1 - 2.24T + 31T^{2} $
	37	$ 1 - 6iT - 37T^{2} $
	41	$ 1 - 11.2T + 41T^{2} $
	43	$ 1 - 5.28iT - 43T^{2} $
	47	$ 1 + 3.45iT - 47T^{2} $
	53	$ 1 + 9.21iT - 53T^{2} $
	59	$ 1 - 5.92T + 59T^{2} $
	61	$ 1 + 5.35T + 61T^{2} $
	67	$ 1 + 7.52iT - 67T^{2} $
	71	$ 1 - 4.24T + 71T^{2} $
	73	$ 1 + 7.28iT - 73T^{2} $
	79	$ 1 - 16.9T + 79T^{2} $
	83	$ 1 - 10.1iT - 83T^{2} $
	89	$ 1 - 11.4T + 89T^{2} $
	97	$ 1 + 2.73iT - 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

−8.585787320300162414984979716790, −7.84283249687865974627752227628, −7.63767507161717843296708045257, −6.50160099248711202574233235904, −6.25331745278574208946108950356, −5.30474464095806988336713891215, −3.71608930648193449067533778337, −2.99206637946372765539977578482, −1.97090035333460657661918828056, −0.914661950823597455975674231028, 0.69051076853526181790528913130, 2.59813123546536225904659346927, 3.74287820290058109988341417975, 4.35840471320251195495591591283, 4.73102071275448978242734808666, 5.68261215996331796987136983009, 6.75560903128466260526524675996, 7.72540096622949161999268205488, 8.774408882708346627620975178610, 9.169561675790777759037865548303

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.464000867\)
\(L(\frac12)\)	\(\approx\)	\(1.464000867\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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