LMFDB - L-function 2-40e2-8.5-c1-0-19

Dirichlet series

L(s) = 1

− 4.47·7^-s + 3·9^-s − 2i·11^-s + 4.47i·13^-s − 6i·19^-s + 4.47·23^-s − 4.47i·37^-s + 2·41^-s + 13.4·47^-s + 13.0·49^-s − 13.4i·53^-s − 14i·59^-s − 13.4·63^-s + 8.94i·77^-s + 9·81^-s + ⋯

L(s) = 1

− 1.69·7^-s + 9^-s − 0.603i·11^-s + 1.24i·13^-s − 1.37i·19^-s + 0.932·23^-s − 0.735i·37^-s + 0.312·41^-s + 1.95·47^-s + 1.85·49^-s − 1.84i·53^-s − 1.82i·59^-s − 1.69·63^-s + 1.01i·77^-s + 81^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1600$ = $2^{6} \cdot 5^{2}$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$12.7760$
Root analytic conductor:	$3.57436$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1600} (801, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1600,\ (\ :1/2),\ 0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.349336861$
$L(\frac12)$	$\approx$	$1.349336861$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
good	3	$ 1 - 3T^{2} $
	7	$ 1 + 4.47T + 7T^{2} $
	11	$ 1 + 2iT - 11T^{2} $
	13	$ 1 - 4.47iT - 13T^{2} $
	17	$ 1 + 17T^{2} $
	19	$ 1 + 6iT - 19T^{2} $
	23	$ 1 - 4.47T + 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 4.47iT - 37T^{2} $
	41	$ 1 - 2T + 41T^{2} $
	43	$ 1 - 43T^{2} $
	47	$ 1 - 13.4T + 47T^{2} $
	53	$ 1 + 13.4iT - 53T^{2} $
	59	$ 1 + 14iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 - 83T^{2} $
	89	$ 1 - 14T + 89T^{2} $
	97	$ 1 + 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

−9.247360714825304157730854366274, −8.883528848959645117931558261086, −7.45892340454149388356168126880, −6.78251091590754585060568078516, −6.36858298557821279831329280348, −5.16233363972040997814095375736, −4.13315366744863959186005773578, −3.34764965365937316480056464511, −2.27036148784765714704029151201, −0.63939751647073664019684694016, 1.06063922479527703628902030604, 2.63193185937014753812212652388, 3.50486703686459883406727326561, 4.35658374271407549268342316965, 5.57962107921070869285111769285, 6.26262096637776380756624938629, 7.18165167170475386855303976401, 7.68730000042798605612201208551, 8.901130875076345026769140171859, 9.601604397945858590463785332239

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.349336861\)
\(L(\frac12)\)	\(\approx\)	\(1.349336861\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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