LMFDB - L-function 2-1152-8.5-c1-0-15

Dirichlet series

L(s) = 1

− 2.82i·11^-s − 6·17^-s − 8.48i·19^-s + 5·25^-s + 6·41^-s − 8.48i·43^-s − 7·49^-s − 14.1i·59^-s − 8.48i·67^-s − 2·73^-s − 2.82i·83^-s + 18·89^-s − 10·97^-s + 19.7i·107^-s − 18·113^-s + ⋯

L(s) = 1

− 0.852i·11^-s − 1.45·17^-s − 1.94i·19^-s + 25^-s + 0.937·41^-s − 1.29i·43^-s − 49^-s − 1.84i·59^-s − 1.03i·67^-s − 0.234·73^-s − 0.310i·83^-s + 1.90·89^-s − 1.01·97^-s + 1.91i·107^-s − 1.69·113^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1152$ = $2^{7} \cdot 3^{2}$
Sign:	$i$
Analytic conductor:	$9.19876$
Root analytic conductor:	$3.03294$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1152} (577, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1152,\ (\ :1/2),\ i)$

Particular Values

$L(1)$	$\approx$	$1.222918354$
$L(\frac12)$	$\approx$	$1.222918354$
$L(\frac{3}{2})$		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 - 5T^{2} $
	7	$ 1 + 7T^{2} $
	11	$ 1 + 2.82iT - 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 + 6T + 17T^{2} $
	19	$ 1 + 8.48iT - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 - 37T^{2} $
	41	$ 1 - 6T + 41T^{2} $
	43	$ 1 + 8.48iT - 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 53T^{2} $
	59	$ 1 + 14.1iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 + 8.48iT - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 2T + 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 + 2.82iT - 83T^{2} $
	89	$ 1 - 18T + 89T^{2} $
	97	$ 1 + 10T + 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

−9.308052110342974313763519651679, −8.934112447492472640506451725280, −8.048121073898811792375318381662, −6.92444938786501759950156107667, −6.42418138180048171022465868752, −5.21262286955756833180502824500, −4.46896829721840085049587143533, −3.24748871615377411163213844306, −2.25745157251685861152483598533, −0.53341778551785973348022687080, 1.53493631811794505241308354975, 2.68995322414667914754180710961, 3.99540873574152028944883734374, 4.72953173899036073328979624469, 5.85988215348902339386597827015, 6.67148895424106404354620094510, 7.53817735829035169437157483481, 8.367577345882093467593828327137, 9.215013384766787839614254997182, 10.00896780047405071802409465561

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.222918354\)
\(L(\frac12)\)	\(\approx\)	\(1.222918354\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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