LMFDB - L-function 2-3330-5.4-c1-0-15

Dirichlet series

L(s) = 1

− i·2^-s − 4^-s + (−2.20 − 0.375i)5^-s − 0.496i·7^-s + i·8^-s + (−0.375 + 2.20i)10^-s − 3.54·11^-s − 3.71i·13^-s − 0.496·14^-s + 16^-s + 6.87i·17^-s − 3.45·19^-s + (2.20 + 0.375i)20^-s + 3.54i·22^-s − 5.34i·23^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s + (−0.985 − 0.168i)5^-s − 0.187i·7^-s + 0.353i·8^-s + (−0.118 + 0.697i)10^-s − 1.06·11^-s − 1.03i·13^-s − 0.132·14^-s + 0.250·16^-s + 1.66i·17^-s − 0.793·19^-s + (0.492 + 0.0840i)20^-s + 0.755i·22^-s − 1.11i·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3330$ = $2 \cdot 3^{2} \cdot 5 \cdot 37$
Sign:	$0.985 + 0.168i$
Analytic conductor:	$26.5901$
Root analytic conductor:	$5.15656$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3330} (1999, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3330,\ (\ :1/2),\ 0.985 + 0.168i)$

Particular Values

$L(1)$	$\approx$	$0.7819822960$
$L(\frac12)$	$\approx$	$0.7819822960$
$L(\frac{3}{2})$		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 + iT $
	3	$ 1 $
	5	$ 1 + (2.20 + 0.375i)T $
	37	$ 1 + iT $
good	7	$ 1 + 0.496iT - 7T^{2} $
	11	$ 1 + 3.54T + 11T^{2} $
	13	$ 1 + 3.71iT - 13T^{2} $
	17	$ 1 - 6.87iT - 17T^{2} $
	19	$ 1 + 3.45T + 19T^{2} $
	23	$ 1 + 5.34iT - 23T^{2} $
	29	$ 1 + 5.00T + 29T^{2} $
	31	$ 1 + 4.05T + 31T^{2} $
	41	$ 1 - 7.78T + 41T^{2} $
	43	$ 1 + 5.91iT - 43T^{2} $
	47	$ 1 - 9.61iT - 47T^{2} $
	53	$ 1 - 12.0iT - 53T^{2} $
	59	$ 1 + 11.4T + 59T^{2} $
	61	$ 1 - 1.18T + 61T^{2} $
	67	$ 1 + 6.56iT - 67T^{2} $
	71	$ 1 - 15.7T + 71T^{2} $
	73	$ 1 - 0.584iT - 73T^{2} $
	79	$ 1 - 1.89T + 79T^{2} $
	83	$ 1 - 3.12iT - 83T^{2} $
	89	$ 1 + 2.39T + 89T^{2} $
	97	$ 1 - 9.01iT - 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

−8.551357179133675847962392421125, −7.919010473322780199074364051322, −7.48695444015155865881739298616, −6.21764754988249392844354390276, −5.47475626733727895427863070472, −4.49433994579989929939899361893, −3.90768770106864266400358218956, −3.04476986914475217172440553486, −2.09443488159916877140745770102, −0.69481376279759242384044803524, 0.37312047718024403425734255468, 2.14438927427003753286369405013, 3.23058830618573428480587771982, 4.10214941258541658256734644711, 4.90943503726677713723792685330, 5.54174687668838056193812862279, 6.60070315546980473493225435236, 7.31349409664342290168087697934, 7.67286383015702841609432692999, 8.542399716967116291059285520474

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(0.7819822960\)
\(L(\frac12)\)	\(\approx\)	\(0.7819822960\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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