LMFDB - L-function 2-2400-40.29-c1-0-3

Dirichlet series

L(s) = 1

− 3^-s − 0.941i·7^-s + 9^-s − 4.49i·11^-s − 5.55·13^-s + 7.55i·17^-s + 1.05i·19^-s + 0.941i·21^-s − 1.05i·23^-s − 27^-s + 2i·29^-s − 3.55·31^-s + 4.49i·33^-s + 7.43·37^-s + 5.55·39^-s + ⋯

L(s) = 1

− 0.577·3^-s − 0.355i·7^-s + 0.333·9^-s − 1.35i·11^-s − 1.54·13^-s + 1.83i·17^-s + 0.242i·19^-s + 0.205i·21^-s − 0.220i·23^-s − 0.192·27^-s + 0.371i·29^-s − 0.638·31^-s + 0.783i·33^-s + 1.22·37^-s + 0.889·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2400$ = $2^{5} \cdot 3 \cdot 5^{2}$
Sign:	$0.271 - 0.962i$
Analytic conductor:	$19.1640$
Root analytic conductor:	$4.37768$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2400} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2400,\ (\ :1/2),\ 0.271 - 0.962i)$

Particular Values

$L(1)$	$\approx$	$0.8527724379$
$L(\frac12)$	$\approx$	$0.8527724379$
$L(\frac{3}{2})$		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 + T $
	5	$ 1 $
good	7	$ 1 + 0.941iT - 7T^{2} $
	11	$ 1 + 4.49iT - 11T^{2} $
	13	$ 1 + 5.55T + 13T^{2} $
	17	$ 1 - 7.55iT - 17T^{2} $
	19	$ 1 - 1.05iT - 19T^{2} $
	23	$ 1 + 1.05iT - 23T^{2} $
	29	$ 1 - 2iT - 29T^{2} $
	31	$ 1 + 3.55T + 31T^{2} $
	37	$ 1 - 7.43T + 37T^{2} $
	41	$ 1 + 3.88T + 41T^{2} $
	43	$ 1 - 1.88T + 43T^{2} $
	47	$ 1 - 10.0iT - 47T^{2} $
	53	$ 1 - 2T + 53T^{2} $
	59	$ 1 + 8.49iT - 59T^{2} $
	61	$ 1 - 8.99iT - 61T^{2} $
	67	$ 1 + 4T + 67T^{2} $
	71	$ 1 - 12.9T + 71T^{2} $
	73	$ 1 - 6iT - 73T^{2} $
	79	$ 1 - 11.5T + 79T^{2} $
	83	$ 1 + 5.88T + 83T^{2} $
	89	$ 1 - 4.11T + 89T^{2} $
	97	$ 1 - 17.1iT - 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

−9.158933388358908501269673163470, −8.229955060372449461510191407196, −7.65036744905544977064513795864, −6.68780689868343760160447035170, −5.99275034026488052956990380863, −5.30259727357758247455658052231, −4.31689890667564807057564405897, −3.52316595033406006892492315126, −2.33764138665036270142862179197, −1.01278378954041542066262668384, 0.36165220567442815438782277080, 2.03294365121442277862368646671, 2.76867761576719492582758553545, 4.21242215406631015143848368256, 4.98272028368521175953375928883, 5.40531479816617333955912510922, 6.66169398789569831672213655676, 7.24924686849625668954259086244, 7.73498500944784953697000767973, 9.067190266583017119875045676000

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8527724379\)
\(L(\frac12)\)	\(\approx\)	\(0.8527724379\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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