LMFDB - L-function 2-3087-21.20-c1-0-51

Dirichlet series

L(s) = 1

− 1.69i·2^-s − 0.867·4^-s + 4.09·5^-s − 1.91i·8^-s − 6.92i·10^-s + 3.45i·11^-s + 1.20i·13^-s − 4.98·16^-s + 4.57·17^-s + 6.51i·19^-s − 3.54·20^-s + 5.85·22^-s + 7.37i·23^-s + 11.7·25^-s + 2.03·26^-s + ⋯

L(s) = 1

− 1.19i·2^-s − 0.433·4^-s + 1.82·5^-s − 0.677i·8^-s − 2.19i·10^-s + 1.04i·11^-s + 0.333i·13^-s − 1.24·16^-s + 1.10·17^-s + 1.49i·19^-s − 0.793·20^-s + 1.24·22^-s + 1.53i·23^-s + 2.34·25^-s + 0.399·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3087$ = $3^{2} \cdot 7^{3}$
Sign:	$0.577 + 0.816i$
Analytic conductor:	$24.6498$
Root analytic conductor:	$4.96485$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3087} (3086, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3087,\ (\ :1/2),\ 0.577 + 0.816i)$

Particular Values

$L(1)$	$\approx$	$2.979116672$
$L(\frac12)$	$\approx$	$2.979116672$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 $
	7	$ 1 $
good	2	$ 1 + 1.69iT - 2T^{2} $
	5	$ 1 - 4.09T + 5T^{2} $
	11	$ 1 - 3.45iT - 11T^{2} $
	13	$ 1 - 1.20iT - 13T^{2} $
	17	$ 1 - 4.57T + 17T^{2} $
	19	$ 1 - 6.51iT - 19T^{2} $
	23	$ 1 - 7.37iT - 23T^{2} $
	29	$ 1 - 3.34iT - 29T^{2} $
	31	$ 1 + 2.85iT - 31T^{2} $
	37	$ 1 - 0.551T + 37T^{2} $
	41	$ 1 - 11.2T + 41T^{2} $
	43	$ 1 - 2.39T + 43T^{2} $
	47	$ 1 + 11.7T + 47T^{2} $
	53	$ 1 - 2.27iT - 53T^{2} $
	59	$ 1 + 0.449T + 59T^{2} $
	61	$ 1 - 3.26iT - 61T^{2} $
	67	$ 1 + 3.57T + 67T^{2} $
	71	$ 1 + 11.6iT - 71T^{2} $
	73	$ 1 + 11.7iT - 73T^{2} $
	79	$ 1 - 9.80T + 79T^{2} $
	83	$ 1 + 13.5T + 83T^{2} $
	89	$ 1 + 15.0T + 89T^{2} $
	97	$ 1 + 3.07iT - 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

−9.130778028219932522828030275181, −7.82268495746685234310600861539, −7.06075479174513585813361053888, −6.10195729171144884126501373445, −5.60408898944388398593368419147, −4.60657691849584990228151586317, −3.56149297015386679670016350039, −2.70558374914265087307932973321, −1.67925107233311618423498764954, −1.47769884686316951138773240036, 0.960933650777431660121892175476, 2.36581264380978755670597026129, 2.93806818876291504350943171554, 4.59121091626315146915510752694, 5.38563992861270279204658329343, 5.85598207249687705220288296541, 6.46599938457878821961291048563, 7.05459441671939277024629408413, 8.120864700734520424481409438446, 8.663123334390574450060103653548

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.979116672\)
\(L(\frac12)\)	\(\approx\)	\(2.979116672\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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