LMFDB - L-function 2-1200-12.11-c1-0-14

Dirichlet series

L(s) = 1

+ 1.73·3^-s + 3.46i·7^-s + 2.99·9^-s + 3.46·11^-s − 4·13^-s + 6i·17^-s + 3.46i·19^-s + 5.99i·21^-s − 3.46·23^-s + 5.19·27^-s − 6i·29^-s − 3.46i·31^-s + 5.99·33^-s + 4·37^-s − 6.92·39^-s + ⋯

L(s) = 1

+ 1.00·3^-s + 1.30i·7^-s + 0.999·9^-s + 1.04·11^-s − 1.10·13^-s + 1.45i·17^-s + 0.794i·19^-s + 1.30i·21^-s − 0.722·23^-s + 1.00·27^-s − 1.11i·29^-s − 0.622i·31^-s + 1.04·33^-s + 0.657·37^-s − 1.10·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1200$ = $2^{4} \cdot 3 \cdot 5^{2}$
Sign:	$0.5 - 0.866i$
Analytic conductor:	$9.58204$
Root analytic conductor:	$3.09548$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1200} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1200,\ (\ :1/2),\ 0.5 - 0.866i)$

Particular Values

$L(1)$	$\approx$	$2.326890053$
$L(\frac12)$	$\approx$	$2.326890053$
$L(\frac{3}{2})$		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 $
	3	$ 1 - 1.73T $
	5	$ 1 $
good	7	$ 1 - 3.46iT - 7T^{2} $
	11	$ 1 - 3.46T + 11T^{2} $
	13	$ 1 + 4T + 13T^{2} $
	17	$ 1 - 6iT - 17T^{2} $
	19	$ 1 - 3.46iT - 19T^{2} $
	23	$ 1 + 3.46T + 23T^{2} $
	29	$ 1 + 6iT - 29T^{2} $
	31	$ 1 + 3.46iT - 31T^{2} $
	37	$ 1 - 4T + 37T^{2} $
	41	$ 1 - 12iT - 41T^{2} $
	43	$ 1 + 6.92iT - 43T^{2} $
	47	$ 1 - 3.46T + 47T^{2} $
	53	$ 1 - 6iT - 53T^{2} $
	59	$ 1 - 3.46T + 59T^{2} $
	61	$ 1 + 10T + 61T^{2} $
	67	$ 1 + 6.92iT - 67T^{2} $
	71	$ 1 - 13.8T + 71T^{2} $
	73	$ 1 - 2T + 73T^{2} $
	79	$ 1 + 10.3iT - 79T^{2} $
	83	$ 1 - 10.3T + 83T^{2} $
	89	$ 1 - 89T^{2} $
	97	$ 1 + 10T + 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

−9.617174337322812066821998858002, −9.154497285542077631355645209698, −8.207093751741031091844427746137, −7.76431221982458566106021140880, −6.46808189778991609006335920607, −5.84430080541105412271687470428, −4.51559807789055526208734608411, −3.70721354746632601359932475889, −2.51901527819013492998931905679, −1.74616850029122072339844568918, 0.927406560044210912211595088998, 2.34015288593240148834715663035, 3.42830021632819486182643434622, 4.27926499727315289089739939820, 5.06278832377350930361764924146, 6.78820023125593017989067828946, 7.11360322846134384945366638025, 7.85498510121115723901522351805, 8.972931571660743922204958213263, 9.509454266858874177720613326992

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.326890053\)
\(L(\frac12)\)	\(\approx\)	\(2.326890053\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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