LMFDB - L-function 2-6030-201.200-c1-0-66

Dirichlet series

L(s) = 1

− 2^-s + 4^-s − 5^-s + 4.73i·7^-s − 8^-s + 10^-s + 5.12·11^-s + 1.74i·13^-s − 4.73i·14^-s + 16^-s − 3.24i·17^-s − 0.725·19^-s − 20^-s − 5.12·22^-s − 4.25i·23^-s + ⋯

L(s) = 1

− 0.707·2^-s + 0.5·4^-s − 0.447·5^-s + 1.79i·7^-s − 0.353·8^-s + 0.316·10^-s + 1.54·11^-s + 0.484i·13^-s − 1.26i·14^-s + 0.250·16^-s − 0.786i·17^-s − 0.166·19^-s − 0.223·20^-s − 1.09·22^-s − 0.886i·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6030$ = $2 \cdot 3^{2} \cdot 5 \cdot 67$
Sign:	$0.144 + 0.989i$
Analytic conductor:	$48.1497$
Root analytic conductor:	$6.93900$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6030} (2411, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6030,\ (\ :1/2),\ 0.144 + 0.989i)$

Particular Values

$L(1)$	$\approx$	$0.7111447867$
$L(\frac12)$	$\approx$	$0.7111447867$
$L(\frac{3}{2})$		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 + T $
	3	$ 1 $
	5	$ 1 + T $
	67	$ 1 + (-5.92 + 5.64i)T $
good	7	$ 1 - 4.73iT - 7T^{2} $
	11	$ 1 - 5.12T + 11T^{2} $
	13	$ 1 - 1.74iT - 13T^{2} $
	17	$ 1 + 3.24iT - 17T^{2} $
	19	$ 1 + 0.725T + 19T^{2} $
	23	$ 1 + 4.25iT - 23T^{2} $
	29	$ 1 + 2.63iT - 29T^{2} $
	31	$ 1 + 8.54iT - 31T^{2} $
	37	$ 1 + 10.8T + 37T^{2} $
	41	$ 1 - 1.43T + 41T^{2} $
	43	$ 1 + 8.70iT - 43T^{2} $
	47	$ 1 - 5.76iT - 47T^{2} $
	53	$ 1 + 6.68T + 53T^{2} $
	59	$ 1 + 4.44iT - 59T^{2} $
	61	$ 1 - 2.57iT - 61T^{2} $
	71	$ 1 + 14.2iT - 71T^{2} $
	73	$ 1 + 10.1T + 73T^{2} $
	79	$ 1 + 3.03iT - 79T^{2} $
	83	$ 1 + 4.51iT - 83T^{2} $
	89	$ 1 - 3.75iT - 89T^{2} $
	97	$ 1 + 10.8iT - 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

−8.122718414342925313505426897746, −7.23811686687958960402537949903, −6.47811684448006760426334573257, −6.04166040921929674318306475949, −5.10785932129947297511390944203, −4.23500217531677494905743004850, −3.29694528216578030014981035581, −2.37480919291760506734575171175, −1.68730492066967026167649884260, −0.25224212406678288562741209745, 1.11697372236349643257063851479, 1.52383858896573741548093114535, 3.22996443204845476017228826496, 3.73118672781660597714630776692, 4.38747607648213012221011371989, 5.42065404911385950733294146250, 6.55264216121623665209583398737, 6.90136852866231568696285158477, 7.48035453442103924499242437002, 8.273641393885718868800855837540

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7111447867\)
\(L(\frac12)\)	\(\approx\)	\(0.7111447867\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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