LMFDB - L-function 2-4600-5.4-c1-0-46

Dirichlet series

L(s) = 1

− 1.23i·3^-s − 0.236i·7^-s + 1.47·9^-s + 11^-s + 2.23i·13^-s + 2.47i·17^-s − 19^-s − 0.291·21^-s − i·23^-s − 5.52i·27^-s + 6.23·29^-s − 8.47·31^-s − 1.23i·33^-s − 6.76i·37^-s + 2.76·39^-s + ⋯

L(s) = 1

− 0.713i·3^-s − 0.0892i·7^-s + 0.490·9^-s + 0.301·11^-s + 0.620i·13^-s + 0.599i·17^-s − 0.229·19^-s − 0.0636·21^-s − 0.208i·23^-s − 1.06i·27^-s + 1.15·29^-s − 1.52·31^-s − 0.215i·33^-s − 1.11i·37^-s + 0.442·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4600$ = $2^{3} \cdot 5^{2} \cdot 23$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$36.7311$
Root analytic conductor:	$6.06062$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4600} (4049, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4600,\ (\ :1/2),\ 0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$2.099200085$
$L(\frac12)$	$\approx$	$2.099200085$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
	23	$ 1 + iT $
good	3	$ 1 + 1.23iT - 3T^{2} $
	7	$ 1 + 0.236iT - 7T^{2} $
	11	$ 1 - T + 11T^{2} $
	13	$ 1 - 2.23iT - 13T^{2} $
	17	$ 1 - 2.47iT - 17T^{2} $
	19	$ 1 + T + 19T^{2} $
	29	$ 1 - 6.23T + 29T^{2} $
	31	$ 1 + 8.47T + 31T^{2} $
	37	$ 1 + 6.76iT - 37T^{2} $
	41	$ 1 - 11.9T + 41T^{2} $
	43	$ 1 - 11.4iT - 43T^{2} $
	47	$ 1 - 1.70iT - 47T^{2} $
	53	$ 1 + 3.23iT - 53T^{2} $
	59	$ 1 - 1.23T + 59T^{2} $
	61	$ 1 + 2.76T + 61T^{2} $
	67	$ 1 - 4.94iT - 67T^{2} $
	71	$ 1 - 10T + 71T^{2} $
	73	$ 1 + 0.527iT - 73T^{2} $
	79	$ 1 - 7.18T + 79T^{2} $
	83	$ 1 - 9iT - 83T^{2} $
	89	$ 1 + 2T + 89T^{2} $
	97	$ 1 - 16.1iT - 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

−8.063375004310325211272214583965, −7.56964521892321194976502745978, −6.75647617338326093496122084601, −6.32019320138899204081364672720, −5.43618571507306981409759320994, −4.34887953236486668901650709436, −3.89867415332916577072895792677, −2.61261581539510535648244937039, −1.79175041630982347049259360433, −0.861547556275380727622899041899, 0.799973499718488600341583529512, 2.05923097985491123177103830134, 3.12389559337084299398196648694, 3.87829425719174037918239165735, 4.63355156205046568612357614609, 5.32588731330798962974952497215, 6.09650644538525204344091050331, 7.05364703761660505664492770827, 7.54720235534033804938995379990, 8.525822970655799763427615386917

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.099200085\)
\(L(\frac12)\)	\(\approx\)	\(2.099200085\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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