LMFDB - L-function 2-1350-5.4-c3-0-40

Dirichlet series

L(s) = 1

− 2i·2^-s − 4·4^-s + 4i·7^-s + 8i·8^-s + 42·11^-s + 20i·13^-s + 8·14^-s + 16·16^-s − 93i·17^-s − 59·19^-s − 84i·22^-s + 9i·23^-s + 40·26^-s − 16i·28^-s − 120·29^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s + 0.215i·7^-s + 0.353i·8^-s + 1.15·11^-s + 0.426i·13^-s + 0.152·14^-s + 0.250·16^-s − 1.32i·17^-s − 0.712·19^-s − 0.814i·22^-s + 0.0815i·23^-s + 0.301·26^-s − 0.107i·28^-s − 0.768·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1350$ = $2 \cdot 3^{3} \cdot 5^{2}$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$79.6525$
Root analytic conductor:	$8.92482$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1350} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1350,\ (\ :3/2),\ 0.447 + 0.894i)$

Particular Values

$L(2)$	$\approx$	$1.983628968$
$L(\frac12)$	$\approx$	$1.983628968$
$L(\frac{5}{2})$		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 + 2iT $
	3	$ 1 $
	5	$ 1 $
good	7	$ 1 - 4iT - 343T^{2} $
	11	$ 1 - 42T + 1.33e3T^{2} $
	13	$ 1 - 20iT - 2.19e3T^{2} $
	17	$ 1 + 93iT - 4.91e3T^{2} $
	19	$ 1 + 59T + 6.85e3T^{2} $
	23	$ 1 - 9iT - 1.21e4T^{2} $
	29	$ 1 + 120T + 2.43e4T^{2} $
	31	$ 1 - 47T + 2.97e4T^{2} $
	37	$ 1 - 262iT - 5.06e4T^{2} $
	41	$ 1 - 126T + 6.89e4T^{2} $
	43	$ 1 + 178iT - 7.95e4T^{2} $
	47	$ 1 + 144iT - 1.03e5T^{2} $
	53	$ 1 - 741iT - 1.48e5T^{2} $
	59	$ 1 - 444T + 2.05e5T^{2} $
	61	$ 1 - 221T + 2.26e5T^{2} $
	67	$ 1 - 538iT - 3.00e5T^{2} $
	71	$ 1 - 690T + 3.57e5T^{2} $
	73	$ 1 + 1.12e3iT - 3.89e5T^{2} $
	79	$ 1 + 665T + 4.93e5T^{2} $
	83	$ 1 - 75iT - 5.71e5T^{2} $
	89	$ 1 - 1.08e3T + 7.04e5T^{2} $
	97	$ 1 + 1.54e3iT - 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

−9.147615805329793846652826920429, −8.630777598559242197788999732232, −7.45305904177883104142022706040, −6.65174236074394373347402543304, −5.68198584630753784983959077845, −4.63651095380745371319392061376, −3.89226455713689136770562952240, −2.81765928628447428332018317483, −1.80384376833010856932727773588, −0.65807779595775183403601786108, 0.78277061714038785617155077545, 2.05021787367763189025816626853, 3.65848701001642783059719922947, 4.17332979981682554490961887896, 5.36727860747056066857200244380, 6.20251906897520799078319469150, 6.81371120112275909901177677097, 7.77779682647120438344187375353, 8.497347995772085792498050979716, 9.210191206923642824138921849973

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.983628968\)
\(L(\frac12)\)	\(\approx\)	\(1.983628968\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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