LMFDB - L-function 2-3200-40.29-c1-0-14

Dirichlet series

L(s) = 1

− 2.23·3^-s + 2.82i·7^-s + 2.00·9^-s − 2.23i·11^-s − 6.32·13^-s − 5i·17^-s − 2.23i·19^-s − 6.32i·21^-s + 5.65i·23^-s + 2.23·27^-s − 6.32i·29^-s + 5.00i·33^-s + 6.32·37^-s + 14.1·39^-s + 3·41^-s + ⋯

L(s) = 1

− 1.29·3^-s + 1.06i·7^-s + 0.666·9^-s − 0.674i·11^-s − 1.75·13^-s − 1.21i·17^-s − 0.512i·19^-s − 1.38i·21^-s + 1.17i·23^-s + 0.430·27^-s − 1.17i·29^-s + 0.870i·33^-s + 1.03·37^-s + 2.26·39^-s + 0.468·41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3200$ = $2^{7} \cdot 5^{2}$
Sign:	$0.447 - 0.894i$
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:	$(2,\ 3200,\ (\ :1/2),\ 0.447 - 0.894i)$

Particular Values

$L(1)$	$\approx$	$0.6362760192$
$L(\frac12)$	$\approx$	$0.6362760192$
$L(\frac{3}{2})$		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 + 2.23T + 3T^{2} $
	7	$ 1 - 2.82iT - 7T^{2} $
	11	$ 1 + 2.23iT - 11T^{2} $
	13	$ 1 + 6.32T + 13T^{2} $
	17	$ 1 + 5iT - 17T^{2} $
	19	$ 1 + 2.23iT - 19T^{2} $
	23	$ 1 - 5.65iT - 23T^{2} $
	29	$ 1 + 6.32iT - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 - 6.32T + 37T^{2} $
	41	$ 1 - 3T + 41T^{2} $
	43	$ 1 - 8.94T + 43T^{2} $
	47	$ 1 - 2.82iT - 47T^{2} $
	53	$ 1 + 12.6T + 53T^{2} $
	59	$ 1 - 8.94iT - 59T^{2} $
	61	$ 1 + 6.32iT - 61T^{2} $
	67	$ 1 + 11.1T + 67T^{2} $
	71	$ 1 + 14.1T + 71T^{2} $
	73	$ 1 + 15iT - 73T^{2} $
	79	$ 1 - 14.1T + 79T^{2} $
	83	$ 1 + 6.70T + 83T^{2} $
	89	$ 1 + T + 89T^{2} $
	97	$ 1 - 10iT - 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

−9.164682270507401831741271956473, −7.77526052302832037500498520258, −7.36404333651414282818571525407, −6.21774970982900809527807989284, −5.83704132985375205997116213277, −5.02742749764096663069953405896, −4.56623238752684872723638007069, −3.00894460282619592593678535657, −2.33440120821207507857513306277, −0.70516200040660348069073472688, 0.36406028847228352297394652487, 1.60966168125923872046140163629, 2.85842460885069053354522728851, 4.29264343856643013661026065421, 4.54821711831870523265548632201, 5.52005721384635720018472587283, 6.26052540772479174785004494437, 7.04716087456950582745593103219, 7.49919806320799352825259882459, 8.424388070197217604442950781230

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6362760192\)
\(L(\frac12)\)	\(\approx\)	\(0.6362760192\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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