LMFDB - L-function 2-12e3-24.11-c3-0-77

Dirichlet series

L(s) = 1

+ 22.3·5^-s + 4.27i·7^-s − 31.4i·11^-s + 374.·25^-s − 218.·29^-s − 327. i·31^-s + 95.4i·35^-s + 324.·49^-s + 756.·53^-s − 703. i·55^-s − 717. i·59^-s + 882.·73^-s + 134.·77^-s − 1.37e3i·79^-s − 621. i·83^-s + ⋯

L(s) = 1

+ 1.99·5^-s + 0.230i·7^-s − 0.862i·11^-s + 2.99·25^-s − 1.39·29^-s − 1.89i·31^-s + 0.460i·35^-s + 0.946·49^-s + 1.96·53^-s − 1.72i·55^-s − 1.58i·59^-s + 1.41·73^-s + 0.199·77^-s − 1.95i·79^-s − 0.821i·83^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$101.955$
Root analytic conductor:	$10.0972$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (863, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :3/2),\ 0.707 + 0.707i)$

Particular Values

$L(2)$	$\approx$	$3.347064178$
$L(\frac12)$	$\approx$	$3.347064178$
$L(\frac{5}{2})$		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 - 22.3T + 125T^{2} $
	7	$ 1 - 4.27iT - 343T^{2} $
	11	$ 1 + 31.4iT - 1.33e3T^{2} $
	13	$ 1 - 2.19e3T^{2} $
	17	$ 1 - 4.91e3T^{2} $
	19	$ 1 + 6.85e3T^{2} $
	23	$ 1 + 1.21e4T^{2} $
	29	$ 1 + 218.T + 2.43e4T^{2} $
	31	$ 1 + 327. iT - 2.97e4T^{2} $
	37	$ 1 - 5.06e4T^{2} $
	41	$ 1 - 6.89e4T^{2} $
	43	$ 1 + 7.95e4T^{2} $
	47	$ 1 + 1.03e5T^{2} $
	53	$ 1 - 756.T + 1.48e5T^{2} $
	59	$ 1 + 717. iT - 2.05e5T^{2} $
	61	$ 1 - 2.26e5T^{2} $
	67	$ 1 + 3.00e5T^{2} $
	71	$ 1 + 3.57e5T^{2} $
	73	$ 1 - 882.T + 3.89e5T^{2} $
	79	$ 1 + 1.37e3iT - 4.93e5T^{2} $
	83	$ 1 + 621. iT - 5.71e5T^{2} $
	89	$ 1 - 7.04e5T^{2} $
	97	$ 1 + 1.29e3T + 9.12e5T^{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

−9.146702887622947734453304283653, −8.220460658540325433026522641402, −7.13484045805436038298151813518, −6.15577786173184079300389117474, −5.77389724445225863799539910802, −5.05494795442239135887800830341, −3.72572419738657238264430956279, −2.54646805351729937467679154065, −1.91250487331663417477359318392, −0.71050355990875859017752679713, 1.16715199747438864154852866651, 1.97983552459157085337686332843, 2.80710971356938535959448214209, 4.14496367670954750401103046440, 5.27835522266571216910300114345, 5.63987197385365639630718872301, 6.76299438392461493329245429546, 7.14614929421418508074034114660, 8.503648738788418373127544067550, 9.211012267022578881359813365171

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(3.347064178\)
\(L(\frac12)\)	\(\approx\)	\(3.347064178\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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