LMFDB - L-function 2-72e2-24.11-c1-0-15

Dirichlet series

L(s) = 1

− 2.24·5^-s + 3.77i·7^-s + 2.57i·11^-s + 2.77i·13^-s + 4.81i·17^-s − 5.77·19^-s + 1.91·23^-s + 0.0289·25^-s + 10.2·29^-s + 6.22i·31^-s − 8.45i·35^-s − 1.96i·37^-s + 10.6i·41^-s − 5.01·43^-s + 6.36·47^-s + ⋯

L(s) = 1

− 1.00·5^-s + 1.42i·7^-s + 0.776i·11^-s + 0.768i·13^-s + 1.16i·17^-s − 1.32·19^-s + 0.398·23^-s + 0.00579·25^-s + 1.90·29^-s + 1.11i·31^-s − 1.42i·35^-s − 0.322i·37^-s + 1.65i·41^-s − 0.764·43^-s + 0.929·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5184$ = $2^{6} \cdot 3^{4}$
Sign:	$-0.965 + 0.258i$
Analytic conductor:	$41.3944$
Root analytic conductor:	$6.43385$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5184} (2591, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5184,\ (\ :1/2),\ -0.965 + 0.258i)$

Particular Values

$L(1)$	$\approx$	$0.8757062089$
$L(\frac12)$	$\approx$	$0.8757062089$
$L(\frac{3}{2})$		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 + 2.24T + 5T^{2} $
	7	$ 1 - 3.77iT - 7T^{2} $
	11	$ 1 - 2.57iT - 11T^{2} $
	13	$ 1 - 2.77iT - 13T^{2} $
	17	$ 1 - 4.81iT - 17T^{2} $
	19	$ 1 + 5.77T + 19T^{2} $
	23	$ 1 - 1.91T + 23T^{2} $
	29	$ 1 - 10.2T + 29T^{2} $
	31	$ 1 - 6.22iT - 31T^{2} $
	37	$ 1 + 1.96iT - 37T^{2} $
	41	$ 1 - 10.6iT - 41T^{2} $
	43	$ 1 + 5.01T + 43T^{2} $
	47	$ 1 - 6.36T + 47T^{2} $
	53	$ 1 - 2.35T + 53T^{2} $
	59	$ 1 - 5.22iT - 59T^{2} $
	61	$ 1 - 2.26iT - 61T^{2} $
	67	$ 1 + 12.5T + 67T^{2} $
	71	$ 1 + 1.19T + 71T^{2} $
	73	$ 1 - 8.57T + 73T^{2} $
	79	$ 1 - 2.39iT - 79T^{2} $
	83	$ 1 + 15.3iT - 83T^{2} $
	89	$ 1 + 14.7iT - 89T^{2} $
	97	$ 1 + 13.7T + 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

−8.612872687426973181956095026580, −8.072412502909322050752039019064, −7.12576943189386028466368588551, −6.44491125432926260183480732792, −5.83330936742570652863904100142, −4.65790038549481387264610904786, −4.39213537580611634686539779444, −3.29296955348504344600238054852, −2.41684673372187280398494894006, −1.54556000217212541492719242446, 0.29884727091823506967234779499, 0.885617150226267974261760259656, 2.51855083538976391079080765278, 3.41287518827263814364349860769, 4.07254907647892239904932110039, 4.67337097295998815010166764725, 5.61295925117878075704164109199, 6.64927376254842924170670385840, 7.08906277781327294205170102298, 7.962301792161702798778335173325

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8757062089\)
\(L(\frac12)\)	\(\approx\)	\(0.8757062089\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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