LMFDB - L-function 2-15e2-3.2-c2-0-0

Dirichlet series

L(s) = 1

− 3.65i·2^-s − 9.32·4^-s − 7.16·7^-s + 19.4i·8^-s + 5.42i·11^-s − 9.81·13^-s + 26.1i·14^-s + 33.6·16^-s + 12.2i·17^-s + 6.32·19^-s + 19.8·22^-s + 12.0i·23^-s + 35.8i·26^-s + 66.7·28^-s − 44.9i·29^-s + ⋯

L(s) = 1

− 1.82i·2^-s − 2.33·4^-s − 1.02·7^-s + 2.42i·8^-s + 0.493i·11^-s − 0.754·13^-s + 1.86i·14^-s + 2.10·16^-s + 0.719i·17^-s + 0.332·19^-s + 0.900·22^-s + 0.523i·23^-s + 1.37i·26^-s + 2.38·28^-s − 1.55i·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$225$ = $3^{2} \cdot 5^{2}$
Sign:	$0.577 - 0.816i$
Analytic conductor:	$6.13080$
Root analytic conductor:	$2.47604$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{225} (26, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 225,\ (\ :1),\ 0.577 - 0.816i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.0142348 + 0.00736849i$
$L(\frac12)$	$\approx$	$0.0142348 + 0.00736849i$
$L(2)$		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 + 3.65iT - 4T^{2} $
	7	$ 1 + 7.16T + 49T^{2} $
	11	$ 1 - 5.42iT - 121T^{2} $
	13	$ 1 + 9.81T + 169T^{2} $
	17	$ 1 - 12.2iT - 289T^{2} $
	19	$ 1 - 6.32T + 361T^{2} $
	23	$ 1 - 12.0iT - 529T^{2} $
	29	$ 1 + 44.9iT - 841T^{2} $
	31	$ 1 + 58.2T + 961T^{2} $
	37	$ 1 + 66.4T + 1.36e3T^{2} $
	41	$ 1 - 16.4iT - 1.68e3T^{2} $
	43	$ 1 - 43.6T + 1.84e3T^{2} $
	47	$ 1 + 40.0iT - 2.20e3T^{2} $
	53	$ 1 + 13.2iT - 2.80e3T^{2} $
	59	$ 1 - 25.1iT - 3.48e3T^{2} $
	61	$ 1 + 35.6T + 3.72e3T^{2} $
	67	$ 1 + 26.7T + 4.48e3T^{2} $
	71	$ 1 - 92.7iT - 5.04e3T^{2} $
	73	$ 1 + 60.3T + 5.32e3T^{2} $
	79	$ 1 + 96.2T + 6.24e3T^{2} $
	83	$ 1 - 79.1iT - 6.88e3T^{2} $
	89	$ 1 + 107. iT - 7.92e3T^{2} $
	97	$ 1 - 1.07T + 9.40e3T^{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

−12.19115001428675673009448748188, −11.25502202083724466382022973649, −10.17658442101766115202544473174, −9.699826623802990102228519336716, −8.763283673038257514687933381052, −7.27232540822787700322405689974, −5.60387729872354185378009938888, −4.21594390502300511314199825844, −3.20340038075304916193212809775, −1.92048620185143657733871544169, 0.008502606129613530832283298733, 3.39183471194962407818964197519, 4.90897585004143669745059241703, 5.82788408101423982083400941443, 6.90105336670096356492159986364, 7.49712896268192577298206420800, 8.866716114407008435009921133579, 9.379940296269984584579710992586, 10.61183876512752258441263325154, 12.30938130838271077698341916555

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

\(L(\frac{3}{2})\)	\(\approx\)	\(0.0142348 + 0.00736849i\)
\(L(\frac12)\)	\(\approx\)	\(0.0142348 + 0.00736849i\)
\(L(2)\)		not available
\(L(1)\)		not available

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