LMFDB - L-function 2-800-5.4-c3-0-0

Dirichlet series

L(s) = 1

+ 10.0i·3^-s − 10.5i·7^-s − 74.9·9^-s − 38.9·11^-s + 68.9i·13^-s + 65.9i·17^-s − 49.4·19^-s + 106.·21^-s + 164. i·23^-s − 484. i·27^-s + 170.·29^-s − 166.·31^-s − 392. i·33^-s − 384. i·37^-s − 696.·39^-s + ⋯

L(s) = 1

+ 1.94i·3^-s − 0.571i·7^-s − 2.77·9^-s − 1.06·11^-s + 1.47i·13^-s + 0.940i·17^-s − 0.597·19^-s + 1.11·21^-s + 1.49i·23^-s − 3.45i·27^-s + 1.09·29^-s − 0.964·31^-s − 2.07i·33^-s − 1.70i·37^-s − 2.85·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$800$ = $2^{5} \cdot 5^{2}$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$47.2015$
Root analytic conductor:	$6.87033$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{800} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 800,\ (\ :3/2),\ 0.447 + 0.894i)$

Particular Values

$L(2)$	$\approx$	$0.1364848321$
$L(\frac12)$	$\approx$	$0.1364848321$
$L(\frac{5}{2})$		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 $
	5	$ 1 $
good	3	$ 1 - 10.0iT - 27T^{2} $
	7	$ 1 + 10.5iT - 343T^{2} $
	11	$ 1 + 38.9T + 1.33e3T^{2} $
	13	$ 1 - 68.9iT - 2.19e3T^{2} $
	17	$ 1 - 65.9iT - 4.91e3T^{2} $
	19	$ 1 + 49.4T + 6.85e3T^{2} $
	23	$ 1 - 164. iT - 1.21e4T^{2} $
	29	$ 1 - 170.T + 2.43e4T^{2} $
	31	$ 1 + 166.T + 2.97e4T^{2} $
	37	$ 1 + 384. iT - 5.06e4T^{2} $
	41	$ 1 + 22.8T + 6.89e4T^{2} $
	43	$ 1 + 136. iT - 7.95e4T^{2} $
	47	$ 1 + 307. iT - 1.03e5T^{2} $
	53	$ 1 + 222iT - 1.48e5T^{2} $
	59	$ 1 - 522.T + 2.05e5T^{2} $
	61	$ 1 - 393.T + 2.26e5T^{2} $
	67	$ 1 + 476. iT - 3.00e5T^{2} $
	71	$ 1 - 4.26T + 3.57e5T^{2} $
	73	$ 1 - 601. iT - 3.89e5T^{2} $
	79	$ 1 + 1.07e3T + 4.93e5T^{2} $
	83	$ 1 + 1.13e3iT - 5.71e5T^{2} $
	89	$ 1 + 479.T + 7.04e5T^{2} $
	97	$ 1 - 635. iT - 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

−10.46745277830568600791542738041, −9.874621189228282794990950421927, −9.021736077295535968874756141516, −8.372519245861047290235036806066, −7.16012425279700964533385260627, −5.85954017285435794965986715145, −5.10353522096865593927578252973, −4.11211803338515966779920695721, −3.64292462135127009382898350380, −2.22510618573877227689657545658, 0.04025275624678756945596929908, 1.02078561666735844725293145893, 2.54713781940208736037664674209, 2.81609764578195789641969138820, 5.02821891680207426208981066282, 5.80211442360034573358066918332, 6.62859732666603899234610119486, 7.48321905928070850486425489674, 8.243893776200507806823524803222, 8.664352014389140131827762309797

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.1364848321\)
\(L(\frac12)\)	\(\approx\)	\(0.1364848321\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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