LMFDB - L-function 2-4600-5.4-c1-0-83

Dirichlet series

L(s) = 1

− 2.61i·3^-s + 3.83i·7^-s − 3.84·9^-s − 0.508·11^-s − 1.01i·13^-s + 1.44i·17^-s + 0.508·19^-s + 10.0·21^-s + i·23^-s + 2.22i·27^-s − 7.51·29^-s − 0.439·31^-s + 1.32i·33^-s − 7.02i·37^-s − 2.64·39^-s + ⋯

L(s) = 1

− 1.51i·3^-s + 1.45i·7^-s − 1.28·9^-s − 0.153·11^-s − 0.280i·13^-s + 0.350i·17^-s + 0.116·19^-s + 2.19·21^-s + 0.208i·23^-s + 0.427i·27^-s − 1.39·29^-s − 0.0788·31^-s + 0.231i·33^-s − 1.15i·37^-s − 0.423·39^-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

$L(1)$	$\approx$	$0.3831350628$
$L(\frac12)$	$\approx$	$0.3831350628$
$L(\frac{3}{2})$		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 + 2.61iT - 3T^{2} $
	7	$ 1 - 3.83iT - 7T^{2} $
	11	$ 1 + 0.508T + 11T^{2} $
	13	$ 1 + 1.01iT - 13T^{2} $
	17	$ 1 - 1.44iT - 17T^{2} $
	19	$ 1 - 0.508T + 19T^{2} $
	29	$ 1 + 7.51T + 29T^{2} $
	31	$ 1 + 0.439T + 31T^{2} $
	37	$ 1 + 7.02iT - 37T^{2} $
	41	$ 1 - 5.47T + 41T^{2} $
	43	$ 1 + 6.72iT - 43T^{2} $
	47	$ 1 + 2.64iT - 47T^{2} $
	53	$ 1 + 4.77iT - 53T^{2} $
	59	$ 1 + 3.85T + 59T^{2} $
	61	$ 1 + 9.05T + 61T^{2} $
	67	$ 1 - 3.45iT - 67T^{2} $
	71	$ 1 + 2.73T + 71T^{2} $
	73	$ 1 - 9.21iT - 73T^{2} $
	79	$ 1 + 10.5T + 79T^{2} $
	83	$ 1 - 1.40iT - 83T^{2} $
	89	$ 1 + 6.77T + 89T^{2} $
	97	$ 1 + 0.313iT - 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

−7.78488772234233743275337850597, −7.27815910920258333067875384572, −6.46835359093520253006390347847, −5.64194967646304781436731137655, −5.47861375486698676888997855864, −4.04079288052389292685402435209, −2.90813989201431834518134226315, −2.21733189626345870890880853643, −1.50795018660753883559115959854, −0.10334653937759163022429664367, 1.34168190487021342366926287444, 2.85835479148872131911889304559, 3.61065980426636772526310121173, 4.33249657671556849369774124443, 4.72061808691258984228118210445, 5.64714433941278087971641587296, 6.53002878295986729402926071466, 7.42244663796806190038572154557, 7.942205201482448352952343779137, 9.034479888665019356397947560952

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.3831350628\)
\(L(\frac12)\)	\(\approx\)	\(0.3831350628\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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