LMFDB - L-function 2-15e2-5.4-c5-0-31

Dirichlet series

L(s) = 1

− 4i·2^-s + 16·4^-s − 225i·7^-s − 192i·8^-s + 434·11^-s − 613i·13^-s − 900·14^-s − 256·16^-s + 878i·17^-s + 731·19^-s − 1.73e3i·22^-s + 2.85e3i·23^-s − 2.45e3·26^-s − 3.60e3i·28^-s − 7.58e3·29^-s + ⋯

L(s) = 1

− 0.707i·2^-s + 0.5·4^-s − 1.73i·7^-s − 1.06i·8^-s + 1.08·11^-s − 1.00i·13^-s − 1.22·14^-s − 0.250·16^-s + 0.736i·17^-s + 0.464·19^-s − 0.764i·22^-s + 1.12i·23^-s − 0.711·26^-s − 0.867i·28^-s − 1.67·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$225$ = $3^{2} \cdot 5^{2}$
Sign:	$-0.894 + 0.447i$
Analytic conductor:	$36.0863$
Root analytic conductor:	$6.00719$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{225} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 225,\ (\ :5/2),\ -0.894 + 0.447i)$

Particular Values

$L(3)$	$\approx$	$2.333831306$
$L(\frac12)$	$\approx$	$2.333831306$
$L(\frac{7}{2})$		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 $
	5	$ 1 $
good	2	$ 1 + 4iT - 32T^{2} $
	7	$ 1 + 225iT - 1.68e4T^{2} $
	11	$ 1 - 434T + 1.61e5T^{2} $
	13	$ 1 + 613iT - 3.71e5T^{2} $
	17	$ 1 - 878iT - 1.41e6T^{2} $
	19	$ 1 - 731T + 2.47e6T^{2} $
	23	$ 1 - 2.85e3iT - 6.43e6T^{2} $
	29	$ 1 + 7.58e3T + 2.05e7T^{2} $
	31	$ 1 - 2.17e3T + 2.86e7T^{2} $
	37	$ 1 + 9.31e3iT - 6.93e7T^{2} $
	41	$ 1 - 1.20e4T + 1.15e8T^{2} $
	43	$ 1 - 1.12e3iT - 1.47e8T^{2} $
	47	$ 1 + 2.98e4iT - 2.29e8T^{2} $
	53	$ 1 - 5.74e3iT - 4.18e8T^{2} $
	59	$ 1 + 5.17e3T + 7.14e8T^{2} $
	61	$ 1 + 3.87e4T + 8.44e8T^{2} $
	67	$ 1 - 3.17e4iT - 1.35e9T^{2} $
	71	$ 1 + 6.44e4T + 1.80e9T^{2} $
	73	$ 1 - 1.97e4iT - 2.07e9T^{2} $
	79	$ 1 - 1.05e5T + 3.07e9T^{2} $
	83	$ 1 + 3.31e3iT - 3.93e9T^{2} $
	89	$ 1 + 6.53e4T + 5.58e9T^{2} $
	97	$ 1 + 8.91e4iT - 8.58e9T^{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

−10.92048885101646033875403853563, −10.26783365073773239997358307839, −9.355562418258033390930461458521, −7.68668666623897820020904613437, −7.09268761161577495451363039028, −5.85899817955017855650964339011, −4.03531330484900295970567952058, −3.42721139637693058686624114420, −1.63860312880416090180511276353, −0.67538304027871857389057474475, 1.76791616078745120525825255003, 2.86171506797837818787058134677, 4.71253525349099420711618876468, 5.91835408564323845345350733703, 6.53865552799869839414018766886, 7.71149413823051440840591205929, 8.903495075609622365856676722944, 9.401574193717058974833673186510, 11.15784565732796546426422114116, 11.77897734978769885806608190057

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

\(L(3)\)	\(\approx\)	\(2.333831306\)
\(L(\frac12)\)	\(\approx\)	\(2.333831306\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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