LMFDB - L-function 2-1200-60.59-c1-0-24

Dirichlet series

L(s) = 1

+ 1.73·3^-s + 3.46·7^-s + 2.99·9^-s + 2i·13^-s − 3.46i·19^-s + 5.99·21^-s + 5.19·27^-s + 10.3i·31^-s − 10i·37^-s + 3.46i·39^-s − 10.3·43^-s + 4.99·49^-s − 5.99i·57^-s + 14·61^-s + 10.3·63^-s + ⋯

L(s) = 1

+ 1.00·3^-s + 1.30·7^-s + 0.999·9^-s + 0.554i·13^-s − 0.794i·19^-s + 1.30·21^-s + 1.00·27^-s + 1.86i·31^-s − 1.64i·37^-s + 0.554i·39^-s − 1.58·43^-s + 0.714·49^-s − 0.794i·57^-s + 1.79·61^-s + 1.30·63^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0599i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1200$ = $2^{4} \cdot 3 \cdot 5^{2}$
Sign:	$0.998 - 0.0599i$
Analytic conductor:	$9.58204$
Root analytic conductor:	$3.09548$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1200} (1199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1200,\ (\ :1/2),\ 0.998 - 0.0599i)$

Particular Values

$L(1)$	$\approx$	$2.761430599$
$L(\frac12)$	$\approx$	$2.761430599$
$L(\frac{3}{2})$		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 - 3.46T + 7T^{2} $
	11	$ 1 + 11T^{2} $
	13	$ 1 - 2iT - 13T^{2} $
	17	$ 1 + 17T^{2} $
	19	$ 1 + 3.46iT - 19T^{2} $
	23	$ 1 - 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 - 10.3iT - 31T^{2} $
	37	$ 1 + 10iT - 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 + 10.3T + 43T^{2} $
	47	$ 1 - 47T^{2} $
	53	$ 1 + 53T^{2} $
	59	$ 1 + 59T^{2} $
	61	$ 1 - 14T + 61T^{2} $
	67	$ 1 + 3.46T + 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 10iT - 73T^{2} $
	79	$ 1 - 17.3iT - 79T^{2} $
	83	$ 1 - 83T^{2} $
	89	$ 1 - 89T^{2} $
	97	$ 1 + 14iT - 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

−9.595451999318768200655447313502, −8.750596246684826980506354968876, −8.316478172809669759893555982684, −7.36267184654783983492866412240, −6.74174307211473500753344745870, −5.25471497313199189338587112581, −4.55899081830842060719394059533, −3.56818881863990235513911504514, −2.36406440546282505852593044709, −1.43179912602284225871614340607, 1.37430379195244422809042879405, 2.37564078097062930691690106768, 3.54688504928705794489292021329, 4.47870421459677721335953946718, 5.34436218789758060566796803941, 6.52929079752262726897329251353, 7.66419592711931657320481990868, 8.072848384797266445732991101890, 8.711216607292374862747768853704, 9.784933607178756571102567335122

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.761430599\)
\(L(\frac12)\)	\(\approx\)	\(2.761430599\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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