LMFDB - L-function 2-5040-105.104-c1-0-64

Dirichlet series

L(s) = 1

− 2.23i·5^-s + (−1.58 + 2.12i)7^-s − 1.41i·11^-s + 3.16·13^-s − 4.47i·17^-s + 6·23^-s − 5.00·25^-s − 2.82i·29^-s + (4.74 + 3.53i)35^-s + 4.24i·37^-s + 9.48·41^-s + 8.48i·43^-s + 4.47i·47^-s + (−1.99 − 6.70i)49^-s − 6·53^-s + ⋯

L(s) = 1

− 0.999i·5^-s + (−0.597 + 0.801i)7^-s − 0.426i·11^-s + 0.877·13^-s − 1.08i·17^-s + 1.25·23^-s − 1.00·25^-s − 0.525i·29^-s + (0.801 + 0.597i)35^-s + 0.697i·37^-s + 1.48·41^-s + 1.29i·43^-s + 0.652i·47^-s + (−0.285 − 0.958i)49^-s − 0.824·53^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0250 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0250 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5040$ = $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$
Sign:	$-0.0250 + 0.999i$
Analytic conductor:	$40.2446$
Root analytic conductor:	$6.34386$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5040} (1889, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5040,\ (\ :1/2),\ -0.0250 + 0.999i)$

Particular Values

$L(1)$	$\approx$	$1.571138668$
$L(\frac12)$	$\approx$	$1.571138668$
$L(\frac{3}{2})$		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 $
	3	$ 1 $
	5	$ 1 + 2.23iT $
	7	$ 1 + (1.58 - 2.12i)T $
good	11	$ 1 + 1.41iT - 11T^{2} $
	13	$ 1 - 3.16T + 13T^{2} $
	17	$ 1 + 4.47iT - 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 - 6T + 23T^{2} $
	29	$ 1 + 2.82iT - 29T^{2} $
	31	$ 1 - 31T^{2} $
	37	$ 1 - 4.24iT - 37T^{2} $
	41	$ 1 - 9.48T + 41T^{2} $
	43	$ 1 - 8.48iT - 43T^{2} $
	47	$ 1 - 4.47iT - 47T^{2} $
	53	$ 1 + 6T + 53T^{2} $
	59	$ 1 + 9.48T + 59T^{2} $
	61	$ 1 + 13.4iT - 61T^{2} $
	67	$ 1 - 67T^{2} $
	71	$ 1 + 5.65iT - 71T^{2} $
	73	$ 1 + 6.32T + 73T^{2} $
	79	$ 1 - 4T + 79T^{2} $
	83	$ 1 + 8.94iT - 83T^{2} $
	89	$ 1 + 9.48T + 89T^{2} $
	97	$ 1 - 12.6T + 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

−8.144132506702445484508512822226, −7.47386786918444063742381446291, −6.33064096580974857441192397206, −6.00351887967457288972069169702, −5.05952789454195759461930693067, −4.53360190632564651329760997313, −3.37815400944779127426623158061, −2.75450030612300859736008839951, −1.50496017323987268878661003682, −0.48644593039133164950878653791, 1.06551986943002071968702390171, 2.22177909883396032332225653666, 3.26217684815086768875138559224, 3.76227282588229410019788834516, 4.56985802559239277515004276227, 5.79865698310958608378890611867, 6.25575574598057149652902981710, 7.17868513474275254267953828204, 7.34337977691488664442090476329, 8.422527041859037775562437279939

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.571138668\)
\(L(\frac12)\)	\(\approx\)	\(1.571138668\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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