LMFDB - L-function 2-4140-69.68-c1-0-12

Dirichlet series

L(s) = 1

+ 5^-s + 0.420i·7^-s − 4.79·11^-s + 0.120·13^-s + 6.14·17^-s + 0.691i·19^-s + (2.21 − 4.25i)23^-s + 25^-s + 2.25i·29^-s − 2.99·31^-s + 0.420i·35^-s + 1.72i·37^-s − 0.604i·41^-s + 7.79i·43^-s − 2.63i·47^-s + ⋯

L(s) = 1

+ 0.447·5^-s + 0.158i·7^-s − 1.44·11^-s + 0.0333·13^-s + 1.49·17^-s + 0.158i·19^-s + (0.461 − 0.887i)23^-s + 0.200·25^-s + 0.417i·29^-s − 0.537·31^-s + 0.0709i·35^-s + 0.284i·37^-s − 0.0944i·41^-s + 1.18i·43^-s − 0.384i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4140$ = $2^{2} \cdot 3^{2} \cdot 5 \cdot 23$
Sign:	$0.990 - 0.135i$
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:	$(2,\ 4140,\ (\ :1/2),\ 0.990 - 0.135i)$

Particular Values

$L(1)$	$\approx$	$1.924481602$
$L(\frac12)$	$\approx$	$1.924481602$
$L(\frac{3}{2})$		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 + (-2.21 + 4.25i)T $
good	7	$ 1 - 0.420iT - 7T^{2} $
	11	$ 1 + 4.79T + 11T^{2} $
	13	$ 1 - 0.120T + 13T^{2} $
	17	$ 1 - 6.14T + 17T^{2} $
	19	$ 1 - 0.691iT - 19T^{2} $
	29	$ 1 - 2.25iT - 29T^{2} $
	31	$ 1 + 2.99T + 31T^{2} $
	37	$ 1 - 1.72iT - 37T^{2} $
	41	$ 1 + 0.604iT - 41T^{2} $
	43	$ 1 - 7.79iT - 43T^{2} $
	47	$ 1 + 2.63iT - 47T^{2} $
	53	$ 1 - 5.37T + 53T^{2} $
	59	$ 1 + 14.4iT - 59T^{2} $
	61	$ 1 + 10.5iT - 61T^{2} $
	67	$ 1 - 4.65iT - 67T^{2} $
	71	$ 1 - 15.5iT - 71T^{2} $
	73	$ 1 - 10.5T + 73T^{2} $
	79	$ 1 + 1.04iT - 79T^{2} $
	83	$ 1 - 15.0T + 83T^{2} $
	89	$ 1 - 10.7T + 89T^{2} $
	97	$ 1 - 11.6iT - 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

−8.248680697784932648679572382019, −7.86598560116193844784412255679, −6.98319180822559481305111513223, −6.17309167143377438798867680808, −5.29651646165646012065324307653, −5.02149112263746864783663388668, −3.72055134541052740706884388717, −2.89138074957718350490446026215, −2.09616517657320001203503075456, −0.812766307442922890231547982267, 0.76438993678789611900077405741, 1.99810713164748012999473732713, 2.90864285312848985482935400953, 3.69157645670001559983978885990, 4.81449684663881782752678531657, 5.54089490982324579003071308701, 5.92370039607960199348265892494, 7.23332806693023973864470934453, 7.51818072574172909684519449156, 8.349060524740823515278419146048

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.924481602\)
\(L(\frac12)\)	\(\approx\)	\(1.924481602\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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