LMFDB - L-function 2-15e2-15.14-c4-0-7

Dirichlet series

L(s) = 1

− 5.40·2^-s + 13.2·4^-s − 16.7i·7^-s + 15.1·8^-s + 193. i·11^-s − 222. i·13^-s + 90.3i·14^-s − 292.·16^-s − 87.8·17^-s + 477.·19^-s − 1.04e3i·22^-s − 627.·23^-s + 1.20e3i·26^-s − 220. i·28^-s − 204. i·29^-s + ⋯

L(s) = 1

− 1.35·2^-s + 0.825·4^-s − 0.341i·7^-s + 0.236·8^-s + 1.60i·11^-s − 1.31i·13^-s + 0.461i·14^-s − 1.14·16^-s − 0.304·17^-s + 1.32·19^-s − 2.16i·22^-s − 1.18·23^-s + 1.77i·26^-s − 0.281i·28^-s − 0.243i·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(5-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$225$ = $3^{2} \cdot 5^{2}$
Sign:	$0.881 - 0.472i$
Analytic conductor:	$23.2582$
Root analytic conductor:	$4.82267$
Motivic weight:	$4$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{225} (224, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 225,\ (\ :2),\ 0.881 - 0.472i)$

Particular Values

$L(\frac{5}{2})$	$\approx$	$0.7803752934$
$L(\frac12)$	$\approx$	$0.7803752934$
$L(3)$		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 $
	5	$ 1 $
good	2	$ 1 + 5.40T + 16T^{2} $
	7	$ 1 + 16.7iT - 2.40e3T^{2} $
	11	$ 1 - 193. iT - 1.46e4T^{2} $
	13	$ 1 + 222. iT - 2.85e4T^{2} $
	17	$ 1 + 87.8T + 8.35e4T^{2} $
	19	$ 1 - 477.T + 1.30e5T^{2} $
	23	$ 1 + 627.T + 2.79e5T^{2} $
	29	$ 1 + 204. iT - 7.07e5T^{2} $
	31	$ 1 + 1.19e3T + 9.23e5T^{2} $
	37	$ 1 + 1.38e3iT - 1.87e6T^{2} $
	41	$ 1 - 1.31e3iT - 2.82e6T^{2} $
	43	$ 1 - 1.49e3iT - 3.41e6T^{2} $
	47	$ 1 - 2.94e3T + 4.87e6T^{2} $
	53	$ 1 - 3.95e3T + 7.89e6T^{2} $
	59	$ 1 - 3.17e3iT - 1.21e7T^{2} $
	61	$ 1 - 6.29e3T + 1.38e7T^{2} $
	67	$ 1 - 7.86e3iT - 2.01e7T^{2} $
	71	$ 1 - 1.16e3iT - 2.54e7T^{2} $
	73	$ 1 - 5.09e3iT - 2.83e7T^{2} $
	79	$ 1 - 5.20e3T + 3.89e7T^{2} $
	83	$ 1 - 3.13e3T + 4.74e7T^{2} $
	89	$ 1 + 1.36e4iT - 6.27e7T^{2} $
	97	$ 1 - 2.07e3iT - 8.85e7T^{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

−11.37076541153719625903151144877, −10.21229784721382995077730712271, −9.882661583322743274049837976904, −8.788926833111949996889872147121, −7.60202388614834058390144914085, −7.24037018622236361752733523688, −5.52151255063697425303116224724, −4.12594836016285084088658368089, −2.25754962255032597620300559656, −0.845099083969140550004914210735, 0.61103461586649896674823298375, 2.04382771808994104705931895532, 3.77130163445004637445901348497, 5.44950264655934384653143949589, 6.70151217867868463244435743818, 7.77938702475127651835819797918, 8.789715846306542231537222626286, 9.271169324058482418417116107385, 10.41226709812686684700646801912, 11.30787409120048014726829750290

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 $\F_{q}$
Belyi maps

\(L(\frac{5}{2})\)	\(\approx\)	\(0.7803752934\)
\(L(\frac12)\)	\(\approx\)	\(0.7803752934\)
\(L(3)\)		not available
\(L(1)\)		not available

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