LMFDB - L-function 2-450-5.4-c3-0-9

Dirichlet series

L(s) = 1

− 2i·2^-s − 4·4^-s − 4i·7^-s + 8i·8^-s − 12·11^-s + 58i·13^-s − 8·14^-s + 16·16^-s − 66i·17^-s + 100·19^-s + 24i·22^-s + 132i·23^-s + 116·26^-s + 16i·28^-s − 90·29^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s − 0.215i·7^-s + 0.353i·8^-s − 0.328·11^-s + 1.23i·13^-s − 0.152·14^-s + 0.250·16^-s − 0.941i·17^-s + 1.20·19^-s + 0.232i·22^-s + 1.19i·23^-s + 0.874·26^-s + 0.107i·28^-s − 0.576·29^-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

$L(2)$	$\approx$	$1.718524662$
$L(\frac12)$	$\approx$	$1.718524662$
$L(\frac{5}{2})$		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 + 12T + 1.33e3T^{2} $
	13	$ 1 - 58iT - 2.19e3T^{2} $
	17	$ 1 + 66iT - 4.91e3T^{2} $
	19	$ 1 - 100T + 6.85e3T^{2} $
	23	$ 1 - 132iT - 1.21e4T^{2} $
	29	$ 1 + 90T + 2.43e4T^{2} $
	31	$ 1 - 152T + 2.97e4T^{2} $
	37	$ 1 + 34iT - 5.06e4T^{2} $
	41	$ 1 - 438T + 6.89e4T^{2} $
	43	$ 1 + 32iT - 7.95e4T^{2} $
	47	$ 1 - 204iT - 1.03e5T^{2} $
	53	$ 1 - 222iT - 1.48e5T^{2} $
	59	$ 1 - 420T + 2.05e5T^{2} $
	61	$ 1 - 902T + 2.26e5T^{2} $
	67	$ 1 + 1.02e3iT - 3.00e5T^{2} $
	71	$ 1 + 432T + 3.57e5T^{2} $
	73	$ 1 + 362iT - 3.89e5T^{2} $
	79	$ 1 - 160T + 4.93e5T^{2} $
	83	$ 1 - 72iT - 5.71e5T^{2} $
	89	$ 1 - 810T + 7.04e5T^{2} $
	97	$ 1 - 1.10e3iT - 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

−10.72848523748022034471888247187, −9.560657420168125583345953404815, −9.241127417908519745188081306337, −7.85307847814056745705920043101, −7.05372382141228285960933022142, −5.68093892312792666205441509627, −4.65756054180351006520017865160, −3.56924481079303792323852415316, −2.34921737102649215019836779592, −0.959846312424796667948307735738, 0.75602300209059065478118762817, 2.67582355966347299025569646735, 3.98117166468712184266240546502, 5.26132311540951957578189144581, 5.93781691166390590797326251258, 7.09416946760370309077869307043, 8.019383443235623041770190810278, 8.671545843867980458318146727698, 9.867074630777899289913721191661, 10.51804494624323660398869690701

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.718524662\)
\(L(\frac12)\)	\(\approx\)	\(1.718524662\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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