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

Dirichlet series

L(s) = 1

− 4.76·3^-s + 16.8i·5^-s − 4.26·9^-s + 40.7i·11^-s − 56.7i·13^-s − 80.5i·15^-s − 122. i·17^-s + 75.4·19^-s − 106. i·23^-s − 160.·25^-s + 149.·27^-s − 146.·29^-s + 42.5·31^-s − 194. i·33^-s + 80.9·37^-s + ⋯

L(s) = 1

− 0.917·3^-s + 1.51i·5^-s − 0.157·9^-s + 1.11i·11^-s − 1.21i·13^-s − 1.38i·15^-s − 1.75i·17^-s + 0.911·19^-s − 0.962i·23^-s − 1.28·25^-s + 1.06·27^-s − 0.939·29^-s + 0.246·31^-s − 1.02i·33^-s + 0.359·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$784$ = $2^{4} \cdot 7^{2}$
Sign:	$0.944 - 0.327i$
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:	$(2,\ 784,\ (\ :3/2),\ 0.944 - 0.327i)$

Particular Values

$L(2)$	$\approx$	$1.146900145$
$L(\frac12)$	$\approx$	$1.146900145$
$L(\frac{5}{2})$		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 + 4.76T + 27T^{2} $
	5	$ 1 - 16.8iT - 125T^{2} $
	11	$ 1 - 40.7iT - 1.33e3T^{2} $
	13	$ 1 + 56.7iT - 2.19e3T^{2} $
	17	$ 1 + 122. iT - 4.91e3T^{2} $
	19	$ 1 - 75.4T + 6.85e3T^{2} $
	23	$ 1 + 106. iT - 1.21e4T^{2} $
	29	$ 1 + 146.T + 2.43e4T^{2} $
	31	$ 1 - 42.5T + 2.97e4T^{2} $
	37	$ 1 - 80.9T + 5.06e4T^{2} $
	41	$ 1 - 53.8iT - 6.89e4T^{2} $
	43	$ 1 - 341. iT - 7.95e4T^{2} $
	47	$ 1 - 4.12T + 1.03e5T^{2} $
	53	$ 1 - 279.T + 1.48e5T^{2} $
	59	$ 1 - 174.T + 2.05e5T^{2} $
	61	$ 1 + 467. iT - 2.26e5T^{2} $
	67	$ 1 + 753. iT - 3.00e5T^{2} $
	71	$ 1 - 669. iT - 3.57e5T^{2} $
	73	$ 1 + 835. iT - 3.89e5T^{2} $
	79	$ 1 - 1.10e3iT - 4.93e5T^{2} $
	83	$ 1 + 552.T + 5.71e5T^{2} $
	89	$ 1 - 122. iT - 7.04e5T^{2} $
	97	$ 1 - 291. iT - 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

−10.10431420760663204929793015363, −9.438895508779303440596602162922, −7.914459096100567625949019035403, −7.19361974036652705197552812861, −6.52363998027358737059825638543, −5.55374187807768497615064258687, −4.75018006116760922871491012048, −3.20994138113759557299512675351, −2.49496298323615690000981164823, −0.56960358216207807276808639931, 0.70327154519702883421232266857, 1.69222448816147821675605662581, 3.61186601940660291716316729121, 4.54770489748745336435564812274, 5.69334251487815039721661505672, 5.83648141245642652256348070866, 7.21840258970836928054225102158, 8.483300684942479815770468165659, 8.798827801516222662985172490386, 9.812773627264119850130483960969

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.146900145\)
\(L(\frac12)\)	\(\approx\)	\(1.146900145\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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