LMFDB - L-function 2-28e2-28.27-c3-0-38

Dirichlet series

L(s) = 1

+ 1.84·3^-s + 15.9i·5^-s − 23.5·9^-s − 65.4i·11^-s + 34.4i·13^-s + 29.5i·15^-s − 50.6i·17^-s + 36.7·19^-s − 196. i·23^-s − 130.·25^-s − 93.5·27^-s + 129.·29^-s + 248.·31^-s − 121. i·33^-s + 105.·37^-s + ⋯

L(s) = 1

+ 0.355·3^-s + 1.42i·5^-s − 0.873·9^-s − 1.79i·11^-s + 0.735i·13^-s + 0.508i·15^-s − 0.722i·17^-s + 0.443·19^-s − 1.77i·23^-s − 1.04·25^-s − 0.666·27^-s + 0.826·29^-s + 1.43·31^-s − 0.638i·33^-s + 0.467·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$784$ = $2^{4} \cdot 7^{2}$
Sign:	$0.933 + 0.358i$
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:	$(2,\ 784,\ (\ :3/2),\ 0.933 + 0.358i)$

Particular Values

$L(2)$	$\approx$	$1.933478997$
$L(\frac12)$	$\approx$	$1.933478997$
$L(\frac{5}{2})$		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 - 1.84T + 27T^{2} $
	5	$ 1 - 15.9iT - 125T^{2} $
	11	$ 1 + 65.4iT - 1.33e3T^{2} $
	13	$ 1 - 34.4iT - 2.19e3T^{2} $
	17	$ 1 + 50.6iT - 4.91e3T^{2} $
	19	$ 1 - 36.7T + 6.85e3T^{2} $
	23	$ 1 + 196. iT - 1.21e4T^{2} $
	29	$ 1 - 129.T + 2.43e4T^{2} $
	31	$ 1 - 248.T + 2.97e4T^{2} $
	37	$ 1 - 105.T + 5.06e4T^{2} $
	41	$ 1 + 186. iT - 6.89e4T^{2} $
	43	$ 1 - 27.7iT - 7.95e4T^{2} $
	47	$ 1 - 15.7T + 1.03e5T^{2} $
	53	$ 1 - 420.T + 1.48e5T^{2} $
	59	$ 1 + 149.T + 2.05e5T^{2} $
	61	$ 1 - 794. iT - 2.26e5T^{2} $
	67	$ 1 - 711. iT - 3.00e5T^{2} $
	71	$ 1 - 798. iT - 3.57e5T^{2} $
	73	$ 1 - 679. iT - 3.89e5T^{2} $
	79	$ 1 + 288. iT - 4.93e5T^{2} $
	83	$ 1 - 1.36e3T + 5.71e5T^{2} $
	89	$ 1 + 913. iT - 7.04e5T^{2} $
	97	$ 1 + 1.77e3iT - 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

−9.988320814468391234788369125633, −8.762137720425569352637650265376, −8.353364333157587712009380141445, −7.13845633376831198356754107642, −6.39787578755092358368754519031, −5.66062677800273406388542748553, −4.17556793532303363757646734111, −2.87737901747484981635720704393, −2.73005231667908312494071639149, −0.59701690618167291765337760542, 0.998293775508219072512715605386, 2.13804145783389048468008723639, 3.49480936457377945296442739202, 4.70130669494629855371227611831, 5.26322196789729372534702386443, 6.37428206441936243779785283925, 7.80374443842724611386048899081, 8.086493968569030371633703832732, 9.252282520105005145750813841134, 9.622829510678958068865614337568

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.933478997\)
\(L(\frac12)\)	\(\approx\)	\(1.933478997\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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