LMFDB - L-function 2-4600-5.4-c1-0-27

Dirichlet series

L(s) = 1

− 3.36i·3^-s + 1.90i·7^-s − 8.28·9^-s − 5.48·11^-s − 1.04i·13^-s + 6.74i·17^-s + 1.55·19^-s + 6.40·21^-s − i·23^-s + 17.7i·27^-s + 3.38·29^-s + 10.9·31^-s + 18.4i·33^-s − 5.26i·37^-s − 3.52·39^-s + ⋯

L(s) = 1

− 1.93i·3^-s + 0.720i·7^-s − 2.76·9^-s − 1.65·11^-s − 0.291i·13^-s + 1.63i·17^-s + 0.355·19^-s + 1.39·21^-s − 0.208i·23^-s + 3.42i·27^-s + 0.628·29^-s + 1.96·31^-s + 3.20i·33^-s − 0.865i·37^-s − 0.564·39^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$4600$ = $2^{3} \cdot 5^{2} \cdot 23$
Sign:	$0.447 + 0.894i$
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:	$(2,\ 4600,\ (\ :1/2),\ 0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$1.326022683$
$L(\frac12)$	$\approx$	$1.326022683$
$L(\frac{3}{2})$		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 + 3.36iT - 3T^{2} $
	7	$ 1 - 1.90iT - 7T^{2} $
	11	$ 1 + 5.48T + 11T^{2} $
	13	$ 1 + 1.04iT - 13T^{2} $
	17	$ 1 - 6.74iT - 17T^{2} $
	19	$ 1 - 1.55T + 19T^{2} $
	29	$ 1 - 3.38T + 29T^{2} $
	31	$ 1 - 10.9T + 31T^{2} $
	37	$ 1 + 5.26iT - 37T^{2} $
	41	$ 1 - 6.09T + 41T^{2} $
	43	$ 1 - 43T^{2} $
	47	$ 1 + 0.403iT - 47T^{2} $
	53	$ 1 - 5.88iT - 53T^{2} $
	59	$ 1 + 9.60T + 59T^{2} $
	61	$ 1 + 7.09T + 61T^{2} $
	67	$ 1 + 13.7iT - 67T^{2} $
	71	$ 1 - 0.478T + 71T^{2} $
	73	$ 1 - 2.40iT - 73T^{2} $
	79	$ 1 - 4.24T + 79T^{2} $
	83	$ 1 + 11.2iT - 83T^{2} $
	89	$ 1 + 4.90T + 89T^{2} $
	97	$ 1 - 12.3iT - 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

−8.023483773802500380438672701928, −7.68354961758836245006548836140, −6.73657885490260994775681118669, −5.95570794922934059607598199005, −5.71780258066526122962758748177, −4.66667543551575697547531484529, −3.10562576866720524314307466914, −2.56699157076202599025529859175, −1.80869219076614632485252519508, −0.68577344853494979427635592576, 0.55996922729282547546059407866, 2.77983141022125751479333583831, 2.93120177236798578475703339484, 4.13929714621833268716539792401, 4.72986573413002767087353325779, 5.17041939636542876034685845224, 5.98757936276875519810773182868, 7.06367327801797111591808747168, 7.950313550903037592950196986886, 8.484476663745345247720153675219

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.326022683\)
\(L(\frac12)\)	\(\approx\)	\(1.326022683\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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