LMFDB - L-function 2-1200-60.59-c1-0-19

Dirichlet series

L(s) = 1

− 1.73·3^-s + 5.19·7^-s + 2.99·9^-s − 7i·13^-s + 5.19i·19^-s − 9·21^-s − 5.19·27^-s + 1.73i·31^-s − 10i·37^-s + 12.1i·39^-s + 1.73·43^-s + 20·49^-s − 9i·57^-s − 61^-s + 15.5·63^-s + ⋯

L(s) = 1

− 1.00·3^-s + 1.96·7^-s + 0.999·9^-s − 1.94i·13^-s + 1.19i·19^-s − 1.96·21^-s − 1.00·27^-s + 0.311i·31^-s − 1.64i·37^-s + 1.94i·39^-s + 0.264·43^-s + 2.85·49^-s − 1.19i·57^-s − 0.128·61^-s + 1.96·63^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1200$ = $2^{4} \cdot 3 \cdot 5^{2}$
Sign:	$0.834 + 0.550i$
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:	$(2,\ 1200,\ (\ :1/2),\ 0.834 + 0.550i)$

Particular Values

$L(1)$	$\approx$	$1.483583614$
$L(\frac12)$	$\approx$	$1.483583614$
$L(\frac{3}{2})$		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 - 5.19T + 7T^{2} $
	11	$ 1 + 11T^{2} $
	13	$ 1 + 7iT - 13T^{2} $
	17	$ 1 + 17T^{2} $
	19	$ 1 - 5.19iT - 19T^{2} $
	23	$ 1 - 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 - 1.73iT - 31T^{2} $
	37	$ 1 + 10iT - 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 - 1.73T + 43T^{2} $
	47	$ 1 - 47T^{2} $
	53	$ 1 + 53T^{2} $
	59	$ 1 + 59T^{2} $
	61	$ 1 + T + 61T^{2} $
	67	$ 1 - 12.1T + 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 - 19iT - 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

−9.984032236899566540939804928529, −8.686561821937392280732477247688, −7.81290632483325185812728639685, −7.48578905327142237613297480225, −6.02219620553493737625539385078, −5.40719083602236903772530540579, −4.77922210016807609782435976090, −3.69742887542673503918349864196, −2.01688487166431710759503963508, −0.883859279042136434844977447775, 1.23862990188887445243971717259, 2.16872630018437580509840248141, 4.20030273483510850580687894976, 4.67522734590156957256078108317, 5.41644707696064774491957438730, 6.59592566059332526814211546733, 7.18761346457029572424150468654, 8.173833551262057705127484575778, 8.990878629567426067556868903106, 9.904006349762734557296986782395

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.483583614\)
\(L(\frac12)\)	\(\approx\)	\(1.483583614\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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