LMFDB - L-function 2-12e2-3.2-c6-0-8

Dirichlet series

L(s) = 1

− 173. i·5^-s + 484·7^-s + 1.34e3i·11^-s + 3.36e3·13^-s − 12.7i·17^-s − 5.74e3·19^-s − 3.37e3i·23^-s − 1.46e4·25^-s − 2.93e4i·29^-s + 3.97e4·31^-s − 8.41e4i·35^-s + 5.25e4·37^-s + 3.70e4i·41^-s − 3.80e3·43^-s + 7.67e4i·47^-s + ⋯

L(s) = 1

− 1.39i·5^-s + 1.41·7^-s + 1.00i·11^-s + 1.53·13^-s − 0.00259i·17^-s − 0.837·19^-s − 0.277i·23^-s − 0.936·25^-s − 1.20i·29^-s + 1.33·31^-s − 1.96i·35^-s + 1.03·37^-s + 0.537i·41^-s − 0.0477·43^-s + 0.739i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$144$ = $2^{4} \cdot 3^{2}$
Sign:	$0.577 + 0.816i$
Analytic conductor:	$33.1277$
Root analytic conductor:	$5.75567$
Motivic weight:	$6$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{144} (17, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 144,\ (\ :3),\ 0.577 + 0.816i)$

Particular Values

$L(\frac{7}{2})$	$\approx$	$2.510495069$
$L(\frac12)$	$\approx$	$2.510495069$
$L(4)$		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 + 173. iT - 1.56e4T^{2} $
	7	$ 1 - 484T + 1.17e5T^{2} $
	11	$ 1 - 1.34e3iT - 1.77e6T^{2} $
	13	$ 1 - 3.36e3T + 4.82e6T^{2} $
	17	$ 1 + 12.7iT - 2.41e7T^{2} $
	19	$ 1 + 5.74e3T + 4.70e7T^{2} $
	23	$ 1 + 3.37e3iT - 1.48e8T^{2} $
	29	$ 1 + 2.93e4iT - 5.94e8T^{2} $
	31	$ 1 - 3.97e4T + 8.87e8T^{2} $
	37	$ 1 - 5.25e4T + 2.56e9T^{2} $
	41	$ 1 - 3.70e4iT - 4.75e9T^{2} $
	43	$ 1 + 3.80e3T + 6.32e9T^{2} $
	47	$ 1 - 7.67e4iT - 1.07e10T^{2} $
	53	$ 1 + 2.38e5iT - 2.21e10T^{2} $
	59	$ 1 + 2.49e5iT - 4.21e10T^{2} $
	61	$ 1 - 1.32e4T + 5.15e10T^{2} $
	67	$ 1 + 1.68e5T + 9.04e10T^{2} $
	71	$ 1 + 5.31e5iT - 1.28e11T^{2} $
	73	$ 1 - 2.36e5T + 1.51e11T^{2} $
	79	$ 1 - 3.51e4T + 2.43e11T^{2} $
	83	$ 1 + 1.09e4iT - 3.26e11T^{2} $
	89	$ 1 - 1.29e5iT - 4.96e11T^{2} $
	97	$ 1 + 3.21e5T + 8.32e11T^{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

−11.83797341295481120918366888217, −10.96299692797315305569050367123, −9.621784142988911707236896353435, −8.443101918666047310890345475419, −8.015378540875718728436061159478, −6.24352548407250608505165201742, −4.87842778730165009840672362270, −4.25270911390618045706173370341, −1.92145876805885855154170684455, −0.916004814440346501894964494509, 1.27129565460516531850455623138, 2.80971070822736746315287972799, 4.07536074456574609125939238883, 5.70754577173580074259172128053, 6.70100945721013363536436576813, 7.982442961510968947010264665721, 8.772853546570716716963485991316, 10.58971707827712798579318381850, 10.94325142230083535338597538769, 11.75658301839717524534462347246

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 $\F_{q}$

\(L(\frac{7}{2})\)	\(\approx\)	\(2.510495069\)
\(L(\frac12)\)	\(\approx\)	\(2.510495069\)
\(L(4)\)		not available
\(L(1)\)		not available

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