LMFDB - L-function 2-40e2-40.29-c1-0-11

Dirichlet series

L(s) = 1

− 1.44·3^-s − 0.898·9^-s + 0.550i·11^-s − 7.89i·17^-s + 8.34i·19^-s + 5.65·27^-s − 0.797i·33^-s + 12.7·41^-s − 10·43^-s + 7·49^-s + 11.4i·51^-s − 12.1i·57^-s + 6i·59^-s + 14.3·67^-s + 13.6i·73^-s + ⋯

L(s) = 1

− 0.836·3^-s − 0.299·9^-s + 0.165i·11^-s − 1.91i·17^-s + 1.91i·19^-s + 1.08·27^-s − 0.138i·33^-s + 1.99·41^-s − 1.52·43^-s + 49^-s + 1.60i·51^-s − 1.60i·57^-s + 0.781i·59^-s + 1.75·67^-s + 1.60i·73^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1600$ = $2^{6} \cdot 5^{2}$
Sign:	$0.948 - 0.316i$
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:	$(2,\ 1600,\ (\ :1/2),\ 0.948 - 0.316i)$

Particular Values

$L(1)$	$\approx$	$1.049988055$
$L(\frac12)$	$\approx$	$1.049988055$
$L(\frac{3}{2})$		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 + 1.44T + 3T^{2} $
	7	$ 1 - 7T^{2} $
	11	$ 1 - 0.550iT - 11T^{2} $
	13	$ 1 + 13T^{2} $
	17	$ 1 + 7.89iT - 17T^{2} $
	19	$ 1 - 8.34iT - 19T^{2} $
	23	$ 1 - 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 37T^{2} $
	41	$ 1 - 12.7T + 41T^{2} $
	43	$ 1 + 10T + 43T^{2} $
	47	$ 1 - 47T^{2} $
	53	$ 1 + 53T^{2} $
	59	$ 1 - 6iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 - 14.3T + 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 - 13.6iT - 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 - 11.4T + 83T^{2} $
	89	$ 1 - 13.8T + 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

−9.584772620088928877764232311187, −8.667108519719853212006780283533, −7.76814689775491676438154745457, −6.99945671631161811272717919968, −6.06018293533426589488756160905, −5.43877429435754413706936530596, −4.62885209241547409491426266975, −3.49025926622925289617435377717, −2.35999444913599819707844510303, −0.821500104886940392774969281301, 0.66800668014292607595188688813, 2.20292753354796729517817336163, 3.40630118259155737007650150794, 4.50242631012420504901707838228, 5.31259092739321913687551925083, 6.16963219705657633235645357219, 6.69676303741450502161001887161, 7.79961179975624945886623842931, 8.619498440208207467402257121171, 9.296000852843757747733622175900

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.049988055\)
\(L(\frac12)\)	\(\approx\)	\(1.049988055\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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