LMFDB - L-function 2-450-5.4-c3-0-3

Dirichlet series

L(s) = 1

− 2i·2^-s − 4·4^-s + i·7^-s + 8i·8^-s − 42·11^-s − 67i·13^-s + 2·14^-s + 16·16^-s + 54i·17^-s + 115·19^-s + 84i·22^-s + 162i·23^-s − 134·26^-s − 4i·28^-s − 210·29^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s + 0.0539i·7^-s + 0.353i·8^-s − 1.15·11^-s − 1.42i·13^-s + 0.0381·14^-s + 0.250·16^-s + 0.770i·17^-s + 1.38·19^-s + 0.814i·22^-s + 1.46i·23^-s − 1.01·26^-s − 0.0269i·28^-s − 1.34·29^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$450$ = $2 \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.447 - 0.894i$
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:	$(2,\ 450,\ (\ :3/2),\ 0.447 - 0.894i)$

Particular Values

$L(2)$	$\approx$	$0.8325663909$
$L(\frac12)$	$\approx$	$0.8325663909$
$L(\frac{5}{2})$		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 - iT - 343T^{2} $
	11	$ 1 + 42T + 1.33e3T^{2} $
	13	$ 1 + 67iT - 2.19e3T^{2} $
	17	$ 1 - 54iT - 4.91e3T^{2} $
	19	$ 1 - 115T + 6.85e3T^{2} $
	23	$ 1 - 162iT - 1.21e4T^{2} $
	29	$ 1 + 210T + 2.43e4T^{2} $
	31	$ 1 + 193T + 2.97e4T^{2} $
	37	$ 1 - 286iT - 5.06e4T^{2} $
	41	$ 1 + 12T + 6.89e4T^{2} $
	43	$ 1 - 263iT - 7.95e4T^{2} $
	47	$ 1 - 414iT - 1.03e5T^{2} $
	53	$ 1 - 192iT - 1.48e5T^{2} $
	59	$ 1 - 690T + 2.05e5T^{2} $
	61	$ 1 + 733T + 2.26e5T^{2} $
	67	$ 1 + 299iT - 3.00e5T^{2} $
	71	$ 1 - 228T + 3.57e5T^{2} $
	73	$ 1 - 938iT - 3.89e5T^{2} $
	79	$ 1 - 160T + 4.93e5T^{2} $
	83	$ 1 - 462iT - 5.71e5T^{2} $
	89	$ 1 + 240T + 7.04e5T^{2} $
	97	$ 1 - 511iT - 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

−10.83873681475319718413332433348, −10.04631242273619391487746896641, −9.276509670919285682082372819313, −8.004239565543575578434790469912, −7.52123912447999566358368013415, −5.72425587523346107784076344787, −5.20584491168773341484575028387, −3.65225842048490600129525126996, −2.78991282878349325069165891270, −1.28978331279074381337272596413, 0.28117832935911331330552939217, 2.20996701283792806053974829948, 3.73581182558786197460205492013, 4.93138856214640589143034796965, 5.71360486807984569434128470588, 7.03886836608338327275433017618, 7.48998783759318748165935589343, 8.739591927397558823403244219767, 9.402775764559550739188966706582, 10.42358364976021129169946794179

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.8325663909\)
\(L(\frac12)\)	\(\approx\)	\(0.8325663909\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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