LMFDB - L-function 2-4600-5.4-c1-0-87

Dirichlet series

L(s) = 1

− 0.724i·3^-s + 2.33i·7^-s + 2.47·9^-s − 2.62·11^-s − 4.29i·13^-s − 6.85i·17^-s − 2.87·19^-s + 1.69·21^-s + i·23^-s − 3.96i·27^-s + 5.03·29^-s − 7.31·31^-s + 1.90i·33^-s + 9.24i·37^-s − 3.11·39^-s + ⋯

L(s) = 1

− 0.418i·3^-s + 0.883i·7^-s + 0.824·9^-s − 0.792·11^-s − 1.19i·13^-s − 1.66i·17^-s − 0.659·19^-s + 0.369·21^-s + 0.208i·23^-s − 0.763i·27^-s + 0.935·29^-s − 1.31·31^-s + 0.331i·33^-s + 1.51i·37^-s − 0.498·39^-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

$L(1)$	$\approx$	$0.7861675252$
$L(\frac12)$	$\approx$	$0.7861675252$
$L(\frac{3}{2})$		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 + 0.724iT - 3T^{2} $
	7	$ 1 - 2.33iT - 7T^{2} $
	11	$ 1 + 2.62T + 11T^{2} $
	13	$ 1 + 4.29iT - 13T^{2} $
	17	$ 1 + 6.85iT - 17T^{2} $
	19	$ 1 + 2.87T + 19T^{2} $
	29	$ 1 - 5.03T + 29T^{2} $
	31	$ 1 + 7.31T + 31T^{2} $
	37	$ 1 - 9.24iT - 37T^{2} $
	41	$ 1 + 6.95T + 41T^{2} $
	43	$ 1 - 7.01iT - 43T^{2} $
	47	$ 1 + 4.74iT - 47T^{2} $
	53	$ 1 + 12.3iT - 53T^{2} $
	59	$ 1 - 2.14T + 59T^{2} $
	61	$ 1 + 2.30T + 61T^{2} $
	67	$ 1 - 1.98iT - 67T^{2} $
	71	$ 1 + 6.87T + 71T^{2} $
	73	$ 1 - 13.2iT - 73T^{2} $
	79	$ 1 + 16.6T + 79T^{2} $
	83	$ 1 - 9.09iT - 83T^{2} $
	89	$ 1 + 0.676T + 89T^{2} $
	97	$ 1 + 15.0iT - 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

−8.184608711534520757624610871620, −7.16130639554780205002507434500, −6.78285341533769777834560158662, −5.64883710720481608155154968285, −5.21391532929184479035799970497, −4.39378948114668665695359884183, −3.10461940884640374115077382064, −2.59278825920438505816928972653, −1.50202874673285258162631824811, −0.20992876490978242350656714206, 1.39448779421601952574778022814, 2.21692322938103891377284903477, 3.60320300653802032491917749404, 4.13194174965719809284695288022, 4.68280456828322761680315553675, 5.71964629216174305773415535120, 6.53008552104614100561351817942, 7.22424954975826214322663317443, 7.78807991341935235032045138242, 8.750874479056850884258132921795

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7861675252\)
\(L(\frac12)\)	\(\approx\)	\(0.7861675252\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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