LMFDB - L-function 2-4140-69.68-c1-0-20

Dirichlet series

L(s) = 1

+ 5^-s − 3.38i·7^-s + 1.93·11^-s + 3.14·13^-s + 0.203·17^-s + 0.518i·19^-s + (0.465 + 4.77i)23^-s + 25^-s + 4.46i·29^-s + 8.42·31^-s − 3.38i·35^-s − 0.490i·37^-s − 3.62i·41^-s + 6.61i·43^-s − 0.426i·47^-s + ⋯

L(s) = 1

+ 0.447·5^-s − 1.28i·7^-s + 0.584·11^-s + 0.870·13^-s + 0.0494·17^-s + 0.118i·19^-s + (0.0970 + 0.995i)23^-s + 0.200·25^-s + 0.829i·29^-s + 1.51·31^-s − 0.572i·35^-s − 0.0805i·37^-s − 0.566i·41^-s + 1.00i·43^-s − 0.0621i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4140$ = $2^{2} \cdot 3^{2} \cdot 5 \cdot 23$
Sign:	$0.868 + 0.495i$
Analytic conductor:	$33.0580$
Root analytic conductor:	$5.74961$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4140} (1241, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4140,\ (\ :1/2),\ 0.868 + 0.495i)$

Particular Values

$L(1)$	$\approx$	$2.389088688$
$L(\frac12)$	$\approx$	$2.389088688$
$L(\frac{3}{2})$		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 - T $
	23	$ 1 + (-0.465 - 4.77i)T $
good	7	$ 1 + 3.38iT - 7T^{2} $
	11	$ 1 - 1.93T + 11T^{2} $
	13	$ 1 - 3.14T + 13T^{2} $
	17	$ 1 - 0.203T + 17T^{2} $
	19	$ 1 - 0.518iT - 19T^{2} $
	29	$ 1 - 4.46iT - 29T^{2} $
	31	$ 1 - 8.42T + 31T^{2} $
	37	$ 1 + 0.490iT - 37T^{2} $
	41	$ 1 + 3.62iT - 41T^{2} $
	43	$ 1 - 6.61iT - 43T^{2} $
	47	$ 1 + 0.426iT - 47T^{2} $
	53	$ 1 - 12.5T + 53T^{2} $
	59	$ 1 - 1.63iT - 59T^{2} $
	61	$ 1 + 13.6iT - 61T^{2} $
	67	$ 1 - 5.74iT - 67T^{2} $
	71	$ 1 - 10.7iT - 71T^{2} $
	73	$ 1 + 8.55T + 73T^{2} $
	79	$ 1 + 9.91iT - 79T^{2} $
	83	$ 1 + 4.71T + 83T^{2} $
	89	$ 1 - 4.48T + 89T^{2} $
	97	$ 1 + 6.61iT - 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

−8.405070782293750986555486401444, −7.52028570670234341682860383460, −6.90923476992628007757783498723, −6.23421431010469258465949702527, −5.44687534636787033895889259687, −4.45334298071460814120874215873, −3.81101346113304045559665948776, −3.00454091265423855263627020037, −1.63125810703527538654883422391, −0.893139943822098322074067562578, 0.988834890833736261265207578503, 2.17341403055673730615905582859, 2.82521853199805303758946037373, 3.92354179139163906817361355754, 4.77650164378577039547203207516, 5.69131738669184220597274944407, 6.19111112980489078213434922109, 6.80332549529887402425778019696, 7.910929175271722388928421176732, 8.761201220652905616978755352160

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.389088688\)
\(L(\frac12)\)	\(\approx\)	\(2.389088688\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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