LMFDB - L-function 2-40e2-40.29-c1-0-18

Dirichlet series

L(s) = 1

− 3^-s + 3.46i·7^-s − 2·9^-s − 3i·11^-s − 3.46·13^-s − 3i·17^-s + i·19^-s − 3.46i·21^-s + 5·27^-s − 10.3i·29^-s + 6.92·31^-s + 3i·33^-s + 10.3·37^-s + 3.46·39^-s − 9·41^-s + ⋯

L(s) = 1

− 0.577·3^-s + 1.30i·7^-s − 0.666·9^-s − 0.904i·11^-s − 0.960·13^-s − 0.727i·17^-s + 0.229i·19^-s − 0.755i·21^-s + 0.962·27^-s − 1.92i·29^-s + 1.24·31^-s + 0.522i·33^-s + 1.70·37^-s + 0.554·39^-s − 1.40·41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1600$ = $2^{6} \cdot 5^{2}$
Sign:	$0.663 + 0.748i$
Analytic conductor:	$12.7760$
Root analytic conductor:	$3.57436$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1600} (1249, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1600,\ (\ :1/2),\ 0.663 + 0.748i)$

Particular Values

$L(1)$	$\approx$	$0.9469744145$
$L(\frac12)$	$\approx$	$0.9469744145$
$L(\frac{3}{2})$		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 + T + 3T^{2} $
	7	$ 1 - 3.46iT - 7T^{2} $
	11	$ 1 + 3iT - 11T^{2} $
	13	$ 1 + 3.46T + 13T^{2} $
	17	$ 1 + 3iT - 17T^{2} $
	19	$ 1 - iT - 19T^{2} $
	23	$ 1 - 23T^{2} $
	29	$ 1 + 10.3iT - 29T^{2} $
	31	$ 1 - 6.92T + 31T^{2} $
	37	$ 1 - 10.3T + 37T^{2} $
	41	$ 1 + 9T + 41T^{2} $
	43	$ 1 - 4T + 43T^{2} $
	47	$ 1 - 10.3iT - 47T^{2} $
	53	$ 1 + 53T^{2} $
	59	$ 1 + 12iT - 59T^{2} $
	61	$ 1 + 3.46iT - 61T^{2} $
	67	$ 1 - 11T + 67T^{2} $
	71	$ 1 - 10.3T + 71T^{2} $
	73	$ 1 - 7iT - 73T^{2} $
	79	$ 1 - 10.3T + 79T^{2} $
	83	$ 1 + 15T + 83T^{2} $
	89	$ 1 - 3T + 89T^{2} $
	97	$ 1 + 14iT - 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

−9.438507946227640046369424483948, −8.375996117283724918355967058943, −7.967538866104044693940380151143, −6.61225455370843592692080664224, −5.95509094023615781303372443677, −5.36111100221311183335086572718, −4.48957355605712433354969311397, −2.98292513368384897572339072330, −2.39152664843649876518415424324, −0.48810051878648358844757487152, 0.982186635103503665799593854644, 2.44521571234295434670274470986, 3.67396254512510274084575324030, 4.64422844565878597205287770010, 5.25729243646382768443004688434, 6.45315831269802301340573313322, 7.02482506751260008589971485695, 7.79174747662742675318894123035, 8.691257436223008386790857807810, 9.723001314476906966876808188399

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9469744145\)
\(L(\frac12)\)	\(\approx\)	\(0.9469744145\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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