LMFDB - L-function 2-28e2-28.27-c3-0-20

Dirichlet series

L(s) = 1

+ 6.34·3^-s + 6.94i·5^-s + 13.3·9^-s − 2.01i·11^-s − 11.3i·13^-s + 44.0i·15^-s + 71.5i·17^-s − 81.8·19^-s + 192. i·23^-s + 76.7·25^-s − 86.9·27^-s − 92.2·29^-s + 252.·31^-s − 12.7i·33^-s + 277.·37^-s + ⋯

L(s) = 1

+ 1.22·3^-s + 0.620i·5^-s + 0.492·9^-s − 0.0551i·11^-s − 0.242i·13^-s + 0.758i·15^-s + 1.02i·17^-s − 0.988·19^-s + 1.74i·23^-s + 0.614·25^-s − 0.619·27^-s − 0.590·29^-s + 1.46·31^-s − 0.0673i·33^-s + 1.23·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$784$ = $2^{4} \cdot 7^{2}$
Sign:	$-0.156 - 0.987i$
Analytic conductor:	$46.2574$
Root analytic conductor:	$6.80128$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{784} (783, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 784,\ (\ :3/2),\ -0.156 - 0.987i)$

Particular Values

$L(2)$	$\approx$	$2.486435347$
$L(\frac12)$	$\approx$	$2.486435347$
$L(\frac{5}{2})$		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 $
	7	$ 1 $
good	3	$ 1 - 6.34T + 27T^{2} $
	5	$ 1 - 6.94iT - 125T^{2} $
	11	$ 1 + 2.01iT - 1.33e3T^{2} $
	13	$ 1 + 11.3iT - 2.19e3T^{2} $
	17	$ 1 - 71.5iT - 4.91e3T^{2} $
	19	$ 1 + 81.8T + 6.85e3T^{2} $
	23	$ 1 - 192. iT - 1.21e4T^{2} $
	29	$ 1 + 92.2T + 2.43e4T^{2} $
	31	$ 1 - 252.T + 2.97e4T^{2} $
	37	$ 1 - 277.T + 5.06e4T^{2} $
	41	$ 1 - 276. iT - 6.89e4T^{2} $
	43	$ 1 - 418. iT - 7.95e4T^{2} $
	47	$ 1 + 416.T + 1.03e5T^{2} $
	53	$ 1 + 310.T + 1.48e5T^{2} $
	59	$ 1 - 184.T + 2.05e5T^{2} $
	61	$ 1 + 148. iT - 2.26e5T^{2} $
	67	$ 1 - 180. iT - 3.00e5T^{2} $
	71	$ 1 + 456. iT - 3.57e5T^{2} $
	73	$ 1 - 874. iT - 3.89e5T^{2} $
	79	$ 1 + 428. iT - 4.93e5T^{2} $
	83	$ 1 + 337.T + 5.71e5T^{2} $
	89	$ 1 + 35.0iT - 7.04e5T^{2} $
	97	$ 1 - 1.60e3iT - 9.12e5T^{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

−9.923857719091759530430357797429, −9.326779711110426297095186416880, −8.192989607003554595682960869381, −7.935690126085468845423060039884, −6.72267868186348102625897144725, −5.89356919604113721979318104932, −4.46889753265332507668342617995, −3.42308620155341407034784771616, −2.71639876453844656605117014562, −1.54017407889677632125207839486, 0.53867382928854658716564209030, 2.09623641115325548763520681422, 2.91270532006465184447051615818, 4.15391565338314936118974156243, 4.92489983450255302719747785343, 6.26723461112898420235747952406, 7.22772705475586651664019226755, 8.283070686441052744406828963269, 8.691728068588941985120417386959, 9.436618062032106776858159494463

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.486435347\)
\(L(\frac12)\)	\(\approx\)	\(2.486435347\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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