LMFDB - L-function 2-4560-76.75-c1-0-42

Dirichlet series

L(s) = 1

− 3^-s + 5^-s − 4.54i·7^-s + 9^-s + 2.47i·11^-s + 2.88i·13^-s − 15^-s + 6.99·17^-s + (3.60 − 2.44i)19^-s + 4.54i·21^-s − 2.99i·23^-s + 25^-s − 27^-s − 0.990i·29^-s + 3.62·31^-s + ⋯

L(s) = 1

− 0.577·3^-s + 0.447·5^-s − 1.71i·7^-s + 0.333·9^-s + 0.745i·11^-s + 0.800i·13^-s − 0.258·15^-s + 1.69·17^-s + (0.827 − 0.561i)19^-s + 0.991i·21^-s − 0.624i·23^-s + 0.200·25^-s − 0.192·27^-s − 0.183i·29^-s + 0.651·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4560$ = $2^{4} \cdot 3 \cdot 5 \cdot 19$
Sign:	$0.899 + 0.436i$
Analytic conductor:	$36.4117$
Root analytic conductor:	$6.03421$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4560} (2431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4560,\ (\ :1/2),\ 0.899 + 0.436i)$

Particular Values

$L(1)$	$\approx$	$1.914810784$
$L(\frac12)$	$\approx$	$1.914810784$
$L(\frac{3}{2})$		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 + T $
	5	$ 1 - T $
	19	$ 1 + (-3.60 + 2.44i)T $
good	7	$ 1 + 4.54iT - 7T^{2} $
	11	$ 1 - 2.47iT - 11T^{2} $
	13	$ 1 - 2.88iT - 13T^{2} $
	17	$ 1 - 6.99T + 17T^{2} $
	23	$ 1 + 2.99iT - 23T^{2} $
	29	$ 1 + 0.990iT - 29T^{2} $
	31	$ 1 - 3.62T + 31T^{2} $
	37	$ 1 - 9.46iT - 37T^{2} $
	41	$ 1 + 1.54iT - 41T^{2} $
	43	$ 1 - 7.98iT - 43T^{2} $
	47	$ 1 + 0.413iT - 47T^{2} $
	53	$ 1 - 6.45iT - 53T^{2} $
	59	$ 1 + 13.0T + 59T^{2} $
	61	$ 1 - 6.65T + 61T^{2} $
	67	$ 1 - 7.16T + 67T^{2} $
	71	$ 1 - 8.62T + 71T^{2} $
	73	$ 1 - 0.592T + 73T^{2} $
	79	$ 1 - 14.9T + 79T^{2} $
	83	$ 1 + 3.02iT - 83T^{2} $
	89	$ 1 + 18.4iT - 89T^{2} $
	97	$ 1 - 11.4iT - 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

−7.937412711970894792736401958309, −7.55466634762688403768664319833, −6.72285762867956935200638230995, −6.32914541110139988483018052403, −5.14211288021543335422498199916, −4.65131002368863770613362036795, −3.86227578078185867201253015662, −2.91340683566798046995550364089, −1.51218324382522046662720764015, −0.840357051893539556461435379470, 0.856873143756245486405282074858, 1.98608776848252407738303607983, 2.99941324105356331760116829138, 3.62875276165047198591815017752, 5.18285771924690455224367976688, 5.54527608382985367117971815046, 5.82009867458390628327804124865, 6.76652826768607699024096827967, 7.87410344621267947239104947950, 8.230551355292577883653582211708

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.914810784\)
\(L(\frac12)\)	\(\approx\)	\(1.914810784\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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