LMFDB - L-function 2-4600-5.4-c1-0-81

Dirichlet series

L(s) = 1

− 2.21i·3^-s − 2.22i·7^-s − 1.88·9^-s − 1.57·11^-s − 3.96i·13^-s + 0.294i·17^-s + 7.76·19^-s − 4.91·21^-s + i·23^-s − 2.46i·27^-s + 9.29·29^-s + 9.18·31^-s + 3.47i·33^-s − 10.5i·37^-s − 8.75·39^-s + ⋯

L(s) = 1

− 1.27i·3^-s − 0.840i·7^-s − 0.628·9^-s − 0.474·11^-s − 1.09i·13^-s + 0.0715i·17^-s + 1.78·19^-s − 1.07·21^-s + 0.208i·23^-s − 0.473i·27^-s + 1.72·29^-s + 1.65·31^-s + 0.605i·33^-s − 1.73i·37^-s − 1.40·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4600$ = $2^{3} \cdot 5^{2} \cdot 23$
Sign:	$-0.894 + 0.447i$
Analytic conductor:	$36.7311$
Root analytic conductor:	$6.06062$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4600} (4049, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4600,\ (\ :1/2),\ -0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$1.984624935$
$L(\frac12)$	$\approx$	$1.984624935$
$L(\frac{3}{2})$		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 $
	23	$ 1 - iT $
good	3	$ 1 + 2.21iT - 3T^{2} $
	7	$ 1 + 2.22iT - 7T^{2} $
	11	$ 1 + 1.57T + 11T^{2} $
	13	$ 1 + 3.96iT - 13T^{2} $
	17	$ 1 - 0.294iT - 17T^{2} $
	19	$ 1 - 7.76T + 19T^{2} $
	29	$ 1 - 9.29T + 29T^{2} $
	31	$ 1 - 9.18T + 31T^{2} $
	37	$ 1 + 10.5iT - 37T^{2} $
	41	$ 1 + 2.34T + 41T^{2} $
	43	$ 1 + 6.67iT - 43T^{2} $
	47	$ 1 + 1.38iT - 47T^{2} $
	53	$ 1 - 11.0iT - 53T^{2} $
	59	$ 1 - 5.09T + 59T^{2} $
	61	$ 1 + 8.91T + 61T^{2} $
	67	$ 1 + 1.12iT - 67T^{2} $
	71	$ 1 - 7.60T + 71T^{2} $
	73	$ 1 + 12.8iT - 73T^{2} $
	79	$ 1 + 11.0T + 79T^{2} $
	83	$ 1 - 13.5iT - 83T^{2} $
	89	$ 1 + 14.3T + 89T^{2} $
	97	$ 1 + 0.199iT - 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

−7.83475127201133874880949115038, −7.35413949987796831142139716774, −6.77384665122417790564311367928, −5.89778796956625589983734802831, −5.22552264452614953331601606754, −4.24770464317464486517642319302, −3.17026074434236672354127518714, −2.49628019861925778036447031836, −1.19047294872526157411443753623, −0.65355185044821512872024189177, 1.29521148969180505669391860748, 2.71208327293603857536999308906, 3.18344941591911917930590773180, 4.36882138665515990858209160524, 4.80572181592264756168484976695, 5.47532537804601658587216701438, 6.38840239415391616693317833932, 7.09367473083844128834969231532, 8.253099927003675265813407154133, 8.592587834978050566186707391527

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.984624935\)
\(L(\frac12)\)	\(\approx\)	\(1.984624935\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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