LMFDB - L-function 2-4608-8.5-c1-0-16

Dirichlet series

L(s) = 1

− 0.449i·5^-s − 2.04·7^-s + 3.46i·11^-s + 4.79·25^-s − 9.34i·29^-s + 3.60·31^-s + 0.921i·35^-s − 2.79·49^-s + 7.55i·53^-s + 1.55·55^-s + 11.3i·59^-s − 9.79·73^-s − 7.10i·77^-s − 17.4·79^-s + 17.3i·83^-s + ⋯

L(s) = 1

− 0.201i·5^-s − 0.774·7^-s + 1.04i·11^-s + 0.959·25^-s − 1.73i·29^-s + 0.647·31^-s + 0.155i·35^-s − 0.399·49^-s + 1.03i·53^-s + 0.209·55^-s + 1.47i·59^-s − 1.14·73^-s − 0.809i·77^-s − 1.96·79^-s + 1.90i·83^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4608$ = $2^{9} \cdot 3^{2}$
Sign:	$-i$
Analytic conductor:	$36.7950$
Root analytic conductor:	$6.06589$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4608} (2305, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4608,\ (\ :1/2),\ -i)$

Particular Values

$L(1)$	$\approx$	$1.161552603$
$L(\frac12)$	$\approx$	$1.161552603$
$L(\frac{3}{2})$		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 $
	3	$ 1 $
good	5	$ 1 + 0.449iT - 5T^{2} $
	7	$ 1 + 2.04T + 7T^{2} $
	11	$ 1 - 3.46iT - 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 + 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 + 9.34iT - 29T^{2} $
	31	$ 1 - 3.60T + 31T^{2} $
	37	$ 1 - 37T^{2} $
	41	$ 1 + 41T^{2} $
	43	$ 1 - 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 7.55iT - 53T^{2} $
	59	$ 1 - 11.3iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 9.79T + 73T^{2} $
	79	$ 1 + 17.4T + 79T^{2} $
	83	$ 1 - 17.3iT - 83T^{2} $
	89	$ 1 + 89T^{2} $
	97	$ 1 + 2T + 97T^{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

−8.536397910515990089512186887165, −7.70728805609599880919121235298, −7.04965236622873560028213392463, −6.35491149615539776981818176087, −5.65001548745027786477061112119, −4.64658104861404087447871988757, −4.13230523946048539733278890043, −3.02764105928208214048831795901, −2.28355486206543831205607822065, −1.03971305660149913197339233163, 0.36249908613628486444621553344, 1.60792618239822217889515284114, 3.06971277623144960839963127037, 3.19804190539352655181826008876, 4.38266443773959193840747956760, 5.26697504229171966403421290887, 6.01800402653040662849709795100, 6.70790913429454865883936693696, 7.24893841972630258272966170416, 8.336209650994975109227913476522

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.161552603\)
\(L(\frac12)\)	\(\approx\)	\(1.161552603\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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