LMFDB - L-function 2-450-5.4-c3-0-18

Dirichlet series

L(s) = 1

− 2i·2^-s − 4·4^-s − 32i·7^-s + 8i·8^-s + 60·11^-s − 34i·13^-s − 64·14^-s + 16·16^-s + 42i·17^-s + 76·19^-s − 120i·22^-s − 68·26^-s + 128i·28^-s + 6·29^-s − 232·31^-s − 32i·32^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s − 1.72i·7^-s + 0.353i·8^-s + 1.64·11^-s − 0.725i·13^-s − 1.22·14^-s + 0.250·16^-s + 0.599i·17^-s + 0.917·19^-s − 1.16i·22^-s − 0.512·26^-s + 0.863i·28^-s + 0.0384·29^-s − 1.34·31^-s − 0.176i·32^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$450$ = $2 \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.894 + 0.447i$
Analytic conductor:	$26.5508$
Root analytic conductor:	$5.15275$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{450} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 450,\ (\ :3/2),\ -0.894 + 0.447i)$

Particular Values

$L(2)$	$\approx$	$1.700499078$
$L(\frac12)$	$\approx$	$1.700499078$
$L(\frac{5}{2})$		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 + 32iT - 343T^{2} $
	11	$ 1 - 60T + 1.33e3T^{2} $
	13	$ 1 + 34iT - 2.19e3T^{2} $
	17	$ 1 - 42iT - 4.91e3T^{2} $
	19	$ 1 - 76T + 6.85e3T^{2} $
	23	$ 1 - 1.21e4T^{2} $
	29	$ 1 - 6T + 2.43e4T^{2} $
	31	$ 1 + 232T + 2.97e4T^{2} $
	37	$ 1 + 134iT - 5.06e4T^{2} $
	41	$ 1 + 234T + 6.89e4T^{2} $
	43	$ 1 + 412iT - 7.95e4T^{2} $
	47	$ 1 + 360iT - 1.03e5T^{2} $
	53	$ 1 + 222iT - 1.48e5T^{2} $
	59	$ 1 - 660T + 2.05e5T^{2} $
	61	$ 1 + 490T + 2.26e5T^{2} $
	67	$ 1 + 812iT - 3.00e5T^{2} $
	71	$ 1 + 120T + 3.57e5T^{2} $
	73	$ 1 - 746iT - 3.89e5T^{2} $
	79	$ 1 + 152T + 4.93e5T^{2} $
	83	$ 1 - 804iT - 5.71e5T^{2} $
	89	$ 1 + 678T + 7.04e5T^{2} $
	97	$ 1 + 194iT - 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

−10.36960486207413218562429654310, −9.635198483667549174820465745977, −8.629838067292762559393991131701, −7.48408825553197541344651142689, −6.71176049185541299580077309044, −5.31839359331918009536537271833, −3.94683224588907942385580563854, −3.57013973419933974095619786190, −1.61124761801604400487030980137, −0.59461831291699184534200897269, 1.56047244359457693387529961113, 3.10864917349973933654939251243, 4.47828883224516753490115609939, 5.56132103344646327153951873948, 6.34031533826960366872582671706, 7.24285252541707020506546643841, 8.504650111796543161488609042482, 9.200758631150008014799649441554, 9.628846829947841016113485089352, 11.41634391908890801855662568750

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.700499078\)
\(L(\frac12)\)	\(\approx\)	\(1.700499078\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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