LMFDB - L-function 2-5520-92.91-c1-0-95

Dirichlet series

L(s) = 1

− i·3^-s − i·5^-s − 3.74·7^-s − 9^-s + 0.951·11^-s − 0.958·13^-s − 15^-s − 7.12i·17^-s − 8.14·19^-s + 3.74i·21^-s + (3.34 − 3.43i)23^-s − 25^-s + i·27^-s + 3.97·29^-s − 10.2i·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 0.447i·5^-s − 1.41·7^-s − 0.333·9^-s + 0.286·11^-s − 0.265·13^-s − 0.258·15^-s − 1.72i·17^-s − 1.86·19^-s + 0.818i·21^-s + (0.697 − 0.716i)23^-s − 0.200·25^-s + 0.192i·27^-s + 0.738·29^-s − 1.84i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5520$ = $2^{4} \cdot 3 \cdot 5 \cdot 23$
Sign:	$-0.272 - 0.962i$
Analytic conductor:	$44.0774$
Root analytic conductor:	$6.63908$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5520} (1471, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5520,\ (\ :1/2),\ -0.272 - 0.962i)$

Particular Values

$L(1)$	$\approx$	$0.1841674883$
$L(\frac12)$	$\approx$	$0.1841674883$
$L(\frac{3}{2})$		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 + iT $
	5	$ 1 + iT $
	23	$ 1 + (-3.34 + 3.43i)T $
good	7	$ 1 + 3.74T + 7T^{2} $
	11	$ 1 - 0.951T + 11T^{2} $
	13	$ 1 + 0.958T + 13T^{2} $
	17	$ 1 + 7.12iT - 17T^{2} $
	19	$ 1 + 8.14T + 19T^{2} $
	29	$ 1 - 3.97T + 29T^{2} $
	31	$ 1 + 10.2iT - 31T^{2} $
	37	$ 1 + 7.97iT - 37T^{2} $
	41	$ 1 - 2.05T + 41T^{2} $
	43	$ 1 - 4.47T + 43T^{2} $
	47	$ 1 - 5.73iT - 47T^{2} $
	53	$ 1 - 5.84iT - 53T^{2} $
	59	$ 1 - 5.24iT - 59T^{2} $
	61	$ 1 - 1.55iT - 61T^{2} $
	67	$ 1 + 8.31T + 67T^{2} $
	71	$ 1 - 14.9iT - 71T^{2} $
	73	$ 1 - 2.66T + 73T^{2} $
	79	$ 1 + 15.4T + 79T^{2} $
	83	$ 1 - 3.86T + 83T^{2} $
	89	$ 1 + 3.08iT - 89T^{2} $
	97	$ 1 + 5.79iT - 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

−7.45614678064388608904870433512, −6.97887920563812808238913337597, −6.22667678988638572632419432262, −5.75076402311436873191600802579, −4.59645564781851457575602369420, −4.02947940886559851186192022925, −2.76322126942181254514241247416, −2.42687420021126207148708566332, −0.867385685733382361409051102809, −0.05854945948714294023653172970, 1.64824236203600840153744239470, 2.80502801055669301654084130976, 3.45408667026583160072483841478, 4.09301622369444806412025894866, 4.97383712472306811075314021200, 6.01465969513600981625198541863, 6.48988734618097643952123094422, 6.94636173425583588035179915036, 8.129220338231806630047037860346, 8.703485770548908919861219920415

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.1841674883\)
\(L(\frac12)\)	\(\approx\)	\(0.1841674883\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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