LMFDB - L-function 2-20e2-20.7-c5-0-25

Dirichlet series

L(s) = 1

− 243i·9^-s + (−475 + 475i)13^-s + (1.52e3 + 1.52e3i)17^-s − 8.56e3i·29^-s + (475 + 475i)37^-s − 4.95e3·41^-s + 1.68e4i·49^-s + (1.64e4 − 1.64e4i)53^-s + 5.49e4·61^-s + (5.44e4 − 5.44e4i)73^-s − 5.90e4·81^-s − 1.40e5i·89^-s + (1.26e5 + 1.26e5i)97^-s + 9.80e4·101^-s − 2.46e5i·109^-s + ⋯

L(s) = 1

− i·9^-s + (−0.779 + 0.779i)13^-s + (1.27 + 1.27i)17^-s − 1.89i·29^-s + (0.0570 + 0.0570i)37^-s − 0.460·41^-s + i·49^-s + (0.805 − 0.805i)53^-s + 1.89·61^-s + (1.19 − 1.19i)73^-s − 0.999·81^-s − 1.87i·89^-s + (1.36 + 1.36i)97^-s + 0.955·101^-s − 1.98i·109^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$400$ = $2^{4} \cdot 5^{2}$
Sign:	$0.850 + 0.525i$
Analytic conductor:	$64.1535$
Root analytic conductor:	$8.00958$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{400} (207, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 400,\ (\ :5/2),\ 0.850 + 0.525i)$

Particular Values

$L(3)$	$\approx$	$1.922895289$
$L(\frac12)$	$\approx$	$1.922895289$
$L(\frac{7}{2})$		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 + 243iT^{2} $
	7	$ 1 - 1.68e4iT^{2} $
	11	$ 1 - 1.61e5T^{2} $
	13	$ 1 + (475 - 475i)T - 3.71e5iT^{2} $
	17	$ 1 + (-1.52e3 - 1.52e3i)T + 1.41e6iT^{2} $
	19	$ 1 + 2.47e6T^{2} $
	23	$ 1 + 6.43e6iT^{2} $
	29	$ 1 + 8.56e3iT - 2.05e7T^{2} $
	31	$ 1 - 2.86e7T^{2} $
	37	$ 1 + (-475 - 475i)T + 6.93e7iT^{2} $
	41	$ 1 + 4.95e3T + 1.15e8T^{2} $
	43	$ 1 + 1.47e8iT^{2} $
	47	$ 1 - 2.29e8iT^{2} $
	53	$ 1 + (-1.64e4 + 1.64e4i)T - 4.18e8iT^{2} $
	59	$ 1 + 7.14e8T^{2} $
	61	$ 1 - 5.49e4T + 8.44e8T^{2} $
	67	$ 1 - 1.35e9iT^{2} $
	71	$ 1 - 1.80e9T^{2} $
	73	$ 1 + (-5.44e4 + 5.44e4i)T - 2.07e9iT^{2} $
	79	$ 1 + 3.07e9T^{2} $
	83	$ 1 + 3.93e9iT^{2} $
	89	$ 1 + 1.40e5iT - 5.58e9T^{2} $
	97	$ 1 + (-1.26e5 - 1.26e5i)T + 8.58e9iT^{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

−10.14352961324209924308368355977, −9.626437047779028774656895765353, −8.549996342411907806984742529465, −7.60133216384385828277995254982, −6.53380084246607686439971068467, −5.70834866217376372879384064520, −4.35518388093080463352917965646, −3.42160442214411884183110652868, −1.99480937220906664094910365709, −0.63185188417247540858696777493, 0.860250705980732214930134167915, 2.35210573542416868815274846457, 3.39055212162405866195969382203, 4.99108206832676203563115094615, 5.42423730894994288232472039899, 7.02072544485713334750403221353, 7.66823918790390842642178553563, 8.639243976711324484461782157148, 9.810105413294988594384792175882, 10.42202382405868765536864122734

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(1.922895289\)
\(L(\frac12)\)	\(\approx\)	\(1.922895289\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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