LMFDB - L-function 2-1350-5.4-c3-0-9

Dirichlet series

L(s) = 1

− 2i·2^-s − 4·4^-s − 8.74i·7^-s + 8i·8^-s − 69.9·11^-s − 6.25i·13^-s − 17.4·14^-s + 16·16^-s − 83.7i·17^-s + 45.7·19^-s + 139. i·22^-s + 56.7i·23^-s − 12.5·26^-s + 34.9i·28^-s − 51.2·29^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s − 0.472i·7^-s + 0.353i·8^-s − 1.91·11^-s − 0.133i·13^-s − 0.333·14^-s + 0.250·16^-s − 1.19i·17^-s + 0.552·19^-s + 1.35i·22^-s + 0.514i·23^-s − 0.0943·26^-s + 0.236i·28^-s − 0.328·29^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$1350$ = $2 \cdot 3^{3} \cdot 5^{2}$
Sign:	$0.894 - 0.447i$
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:	$(2,\ 1350,\ (\ :3/2),\ 0.894 - 0.447i)$

Particular Values

$L(2)$	$\approx$	$0.7470366118$
$L(\frac12)$	$\approx$	$0.7470366118$
$L(\frac{5}{2})$		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 + 8.74iT - 343T^{2} $
	11	$ 1 + 69.9T + 1.33e3T^{2} $
	13	$ 1 + 6.25iT - 2.19e3T^{2} $
	17	$ 1 + 83.7iT - 4.91e3T^{2} $
	19	$ 1 - 45.7T + 6.85e3T^{2} $
	23	$ 1 - 56.7iT - 1.21e4T^{2} $
	29	$ 1 + 51.2T + 2.43e4T^{2} $
	31	$ 1 + 284.T + 2.97e4T^{2} $
	37	$ 1 + 21.2iT - 5.06e4T^{2} $
	41	$ 1 - 14.9T + 6.89e4T^{2} $
	43	$ 1 + 373. iT - 7.95e4T^{2} $
	47	$ 1 - 380. iT - 1.03e5T^{2} $
	53	$ 1 - 54.2iT - 1.48e5T^{2} $
	59	$ 1 - 402.T + 2.05e5T^{2} $
	61	$ 1 - 686.T + 2.26e5T^{2} $
	67	$ 1 - 818. iT - 3.00e5T^{2} $
	71	$ 1 + 596.T + 3.57e5T^{2} $
	73	$ 1 + 454. iT - 3.89e5T^{2} $
	79	$ 1 - 16.7T + 4.93e5T^{2} $
	83	$ 1 - 816. iT - 5.71e5T^{2} $
	89	$ 1 + 663.T + 7.04e5T^{2} $
	97	$ 1 - 817. 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

−9.493021421137996064774175492359, −8.587715367314722693504392961946, −7.57532607481870657279921363346, −7.20940036100852048315662603302, −5.53364398239401196436601487829, −5.20493004442053299892489366761, −4.01160478525048539868546696570, −3.02100465217262171327582642561, −2.21025562861344702280235881136, −0.78421915706834506173353147011, 0.22500041133449366137978317450, 1.92752224176305259088695975860, 3.03713472599943295631638837846, 4.17462960400919434729681699942, 5.33176361236281437792494842776, 5.64307465663977156728427069513, 6.76267694313786886484648566849, 7.64752621149068454974015572436, 8.237201504500786927656097759468, 8.972491972936025619998746851462

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.7470366118\)
\(L(\frac12)\)	\(\approx\)	\(0.7470366118\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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