LMFDB - L-function 2-3200-40.29-c1-0-31

Dirichlet series

L(s) = 1

− 3.16·3^-s + 3.16i·7^-s + 7.00·9^-s − 6·13^-s − 2i·17^-s + 6.32i·19^-s − 10.0i·21^-s − 3.16i·23^-s − 12.6·27^-s − 4i·29^-s − 6.32·31^-s − 2·37^-s + 18.9·39^-s − 3.16·43^-s + 9.48i·47^-s + ⋯

L(s) = 1

− 1.82·3^-s + 1.19i·7^-s + 2.33·9^-s − 1.66·13^-s − 0.485i·17^-s + 1.45i·19^-s − 2.18i·21^-s − 0.659i·23^-s − 2.43·27^-s − 0.742i·29^-s − 1.13·31^-s − 0.328·37^-s + 3.03·39^-s − 0.482·43^-s + 1.38i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3200$ = $2^{7} \cdot 5^{2}$
Sign:	$0.316 + 0.948i$
Analytic conductor:	$25.5521$
Root analytic conductor:	$5.05491$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3200} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3200,\ (\ :1/2),\ 0.316 + 0.948i)$

Particular Values

$L(1)$	$\approx$	$0.2756323663$
$L(\frac12)$	$\approx$	$0.2756323663$
$L(\frac{3}{2})$		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 + 3.16T + 3T^{2} $
	7	$ 1 - 3.16iT - 7T^{2} $
	11	$ 1 - 11T^{2} $
	13	$ 1 + 6T + 13T^{2} $
	17	$ 1 + 2iT - 17T^{2} $
	19	$ 1 - 6.32iT - 19T^{2} $
	23	$ 1 + 3.16iT - 23T^{2} $
	29	$ 1 + 4iT - 29T^{2} $
	31	$ 1 + 6.32T + 31T^{2} $
	37	$ 1 + 2T + 37T^{2} $
	41	$ 1 + 41T^{2} $
	43	$ 1 + 3.16T + 43T^{2} $
	47	$ 1 - 9.48iT - 47T^{2} $
	53	$ 1 - 6T + 53T^{2} $
	59	$ 1 - 6.32iT - 59T^{2} $
	61	$ 1 + 2iT - 61T^{2} $
	67	$ 1 + 9.48T + 67T^{2} $
	71	$ 1 - 6.32T + 71T^{2} $
	73	$ 1 - 14iT - 73T^{2} $
	79	$ 1 + 12.6T + 79T^{2} $
	83	$ 1 - 3.16T + 83T^{2} $
	89	$ 1 + 10T + 89T^{2} $
	97	$ 1 - 2iT - 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

−8.510671266817330053617301476218, −7.49365255762116738947090104069, −6.94424035232873293536403941206, −5.92377450551764237882176955183, −5.64206103939199721689777334376, −4.89380040197586027475804622053, −4.16762997478889683838020378675, −2.69403275424594237689008416250, −1.67405927939902716173914448227, −0.16575872163652090092810194550, 0.71525530113424171503805327399, 1.93348791410992240557218405519, 3.50537414092495214041707404591, 4.49655800217127092750191190580, 4.99941347881160535230578600030, 5.65400879175523940631193450767, 6.74135704230430676719143564455, 7.10913121124760308494959072202, 7.60414841922958193683366356492, 8.997179097410082821983813209599

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.2756323663\)
\(L(\frac12)\)	\(\approx\)	\(0.2756323663\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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