LMFDB - L-function 2-2376-33.32-c1-0-14

Dirichlet series

L(s) = 1

− 1.77i·5^-s + 3.46i·7^-s + (−2.97 + 1.45i)11^-s − 4.16i·13^-s − 4.55·17^-s − 4.38i·19^-s + 6.33i·23^-s + 1.85·25^-s + 6.30·29^-s + 9.22·31^-s + 6.14·35^-s + 11.6·37^-s − 0.120·41^-s + 7.26i·43^-s + 11.4i·47^-s + ⋯

L(s) = 1

− 0.793i·5^-s + 1.30i·7^-s + (−0.898 + 0.439i)11^-s − 1.15i·13^-s − 1.10·17^-s − 1.00i·19^-s + 1.32i·23^-s + 0.370·25^-s + 1.17·29^-s + 1.65·31^-s + 1.03·35^-s + 1.92·37^-s − 0.0187·41^-s + 1.10i·43^-s + 1.66i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.439i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2376$ = $2^{3} \cdot 3^{3} \cdot 11$
Sign:	$0.898 - 0.439i$
Analytic conductor:	$18.9724$
Root analytic conductor:	$4.35573$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2376} (593, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2376,\ (\ :1/2),\ 0.898 - 0.439i)$

Particular Values

$L(1)$	$\approx$	$1.553554563$
$L(\frac12)$	$\approx$	$1.553554563$
$L(\frac{3}{2})$		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 $
	11	$ 1 + (2.97 - 1.45i)T $
good	5	$ 1 + 1.77iT - 5T^{2} $
	7	$ 1 - 3.46iT - 7T^{2} $
	13	$ 1 + 4.16iT - 13T^{2} $
	17	$ 1 + 4.55T + 17T^{2} $
	19	$ 1 + 4.38iT - 19T^{2} $
	23	$ 1 - 6.33iT - 23T^{2} $
	29	$ 1 - 6.30T + 29T^{2} $
	31	$ 1 - 9.22T + 31T^{2} $
	37	$ 1 - 11.6T + 37T^{2} $
	41	$ 1 + 0.120T + 41T^{2} $
	43	$ 1 - 7.26iT - 43T^{2} $
	47	$ 1 - 11.4iT - 47T^{2} $
	53	$ 1 + 6.72iT - 53T^{2} $
	59	$ 1 + 1.12iT - 59T^{2} $
	61	$ 1 - 4.60iT - 61T^{2} $
	67	$ 1 + 5.85T + 67T^{2} $
	71	$ 1 + 14.3iT - 71T^{2} $
	73	$ 1 - 5.41iT - 73T^{2} $
	79	$ 1 - 10.1iT - 79T^{2} $
	83	$ 1 - 10.2T + 83T^{2} $
	89	$ 1 - 11.2iT - 89T^{2} $
	97	$ 1 - 12.5T + 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

−9.077960837655907641874661603357, −8.194178766687238785488356402093, −7.82396825430649757864532540145, −6.56014371343937396152112597356, −5.83079100033196782327876052898, −4.94471517565430569182303861978, −4.60633359003135395961402535210, −2.91569305536490535700681115796, −2.46899153158979878265208956684, −0.938556086562890567480138307312, 0.67657291048216608098798192002, 2.21472594104177305798360213819, 3.07613938732711940661291470570, 4.23271152184569813472599560687, 4.62332860389756877563243713798, 6.07528060026773072967255388750, 6.66656112991957547180644498923, 7.23934774652996843475006003308, 8.154766185017321891527170763855, 8.759370058612323696087992463279

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(1.553554563\)
\(L(\frac12)\)	\(\approx\)	\(1.553554563\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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