LMFDB - L-function 2-1200-5.4-c3-0-38

Dirichlet series

L(s) = 1

− 3i·3^-s + 30.4i·7^-s − 9·9^-s − 31.4·11^-s − 60.7i·13^-s + 121. i·17^-s − 14.4·19^-s + 91.3·21^-s − 13.6i·23^-s + 27i·27^-s + 76.0·29^-s − 183.·31^-s + 94.3i·33^-s + 37.3i·37^-s − 182.·39^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 1.64i·7^-s − 0.333·9^-s − 0.861·11^-s − 1.29i·13^-s + 1.72i·17^-s − 0.174·19^-s + 0.948·21^-s − 0.124i·23^-s + 0.192i·27^-s + 0.486·29^-s − 1.06·31^-s + 0.497i·33^-s + 0.166i·37^-s − 0.748·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1200$ = $2^{4} \cdot 3 \cdot 5^{2}$
Sign:	$-0.447 + 0.894i$
Analytic conductor:	$70.8022$
Root analytic conductor:	$8.41440$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1200} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1200,\ (\ :3/2),\ -0.447 + 0.894i)$

Particular Values

$L(2)$	$\approx$	$0.7740546965$
$L(\frac12)$	$\approx$	$0.7740546965$
$L(\frac{5}{2})$		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 + 3iT $
	5	$ 1 $
good	7	$ 1 - 30.4iT - 343T^{2} $
	11	$ 1 + 31.4T + 1.33e3T^{2} $
	13	$ 1 + 60.7iT - 2.19e3T^{2} $
	17	$ 1 - 121. iT - 4.91e3T^{2} $
	19	$ 1 + 14.4T + 6.85e3T^{2} $
	23	$ 1 + 13.6iT - 1.21e4T^{2} $
	29	$ 1 - 76.0T + 2.43e4T^{2} $
	31	$ 1 + 183.T + 2.97e4T^{2} $
	37	$ 1 - 37.3iT - 5.06e4T^{2} $
	41	$ 1 + 30.6T + 6.89e4T^{2} $
	43	$ 1 + 327. iT - 7.95e4T^{2} $
	47	$ 1 + 449. iT - 1.03e5T^{2} $
	53	$ 1 + 301. iT - 1.48e5T^{2} $
	59	$ 1 - 340.T + 2.05e5T^{2} $
	61	$ 1 - 619.T + 2.26e5T^{2} $
	67	$ 1 - 256. iT - 3.00e5T^{2} $
	71	$ 1 + 499.T + 3.57e5T^{2} $
	73	$ 1 + 19.1iT - 3.89e5T^{2} $
	79	$ 1 - 257.T + 4.93e5T^{2} $
	83	$ 1 + 914. iT - 5.71e5T^{2} $
	89	$ 1 - 1.05e3T + 7.04e5T^{2} $
	97	$ 1 + 521iT - 9.12e5T^{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

−8.688443324897789919627174523650, −8.471540862950880330055114190940, −7.61922851519277336336559594743, −6.47793294598757004775987116402, −5.62459689528988808398976218116, −5.26065697335991506979567257351, −3.62215832498174715950786030892, −2.59967380986998907709268947022, −1.82892265725929209158952661518, −0.20268246308007988490693348643, 0.999963127928804309696004217790, 2.52036886696096643569050649872, 3.65255597212903549214591927310, 4.48862698021965612711391581300, 5.11118388542415773713891450225, 6.44774232635572910016293750434, 7.25256390489212099672420206790, 7.82376582681091171315405150230, 9.053650507171187956988293391085, 9.665497956773840427033866710678

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.7740546965\)
\(L(\frac12)\)	\(\approx\)	\(0.7740546965\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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