LMFDB - L-function 2-45e2-5.4-c1-0-28

Dirichlet series

L(s) = 1

− 2.51i·2^-s − 4.32·4^-s − 0.514i·7^-s + 5.83i·8^-s + 3.32·11^-s + 1.32i·13^-s − 1.29·14^-s + 6.02·16^-s + 3.32i·17^-s + 1.32·19^-s − 8.34i·22^-s + 4.12i·23^-s + 3.32·26^-s + 2.22i·28^-s − 1.38·29^-s + ⋯

L(s) = 1

− 1.77i·2^-s − 2.16·4^-s − 0.194i·7^-s + 2.06i·8^-s + 1.00·11^-s + 0.366i·13^-s − 0.345·14^-s + 1.50·16^-s + 0.805i·17^-s + 0.303·19^-s − 1.78i·22^-s + 0.860i·23^-s + 0.651·26^-s + 0.419i·28^-s − 0.257·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2025$ = $3^{4} \cdot 5^{2}$
Sign:	$-0.447 + 0.894i$
Analytic conductor:	$16.1697$
Root analytic conductor:	$4.02115$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2025} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2025,\ (\ :1/2),\ -0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$1.594630561$
$L(\frac12)$	$\approx$	$1.594630561$
$L(\frac{3}{2})$		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 + 2.51iT - 2T^{2} $
	7	$ 1 + 0.514iT - 7T^{2} $
	11	$ 1 - 3.32T + 11T^{2} $
	13	$ 1 - 1.32iT - 13T^{2} $
	17	$ 1 - 3.32iT - 17T^{2} $
	19	$ 1 - 1.32T + 19T^{2} $
	23	$ 1 - 4.12iT - 23T^{2} $
	29	$ 1 + 1.38T + 29T^{2} $
	31	$ 1 - 8.73T + 31T^{2} $
	37	$ 1 - 0.292iT - 37T^{2} $
	41	$ 1 - 11.3T + 41T^{2} $
	43	$ 1 + 10.3iT - 43T^{2} $
	47	$ 1 - 4.86iT - 47T^{2} $
	53	$ 1 + 5.02iT - 53T^{2} $
	59	$ 1 + 5.02T + 59T^{2} $
	61	$ 1 - 7.34T + 61T^{2} $
	67	$ 1 - 9.44iT - 67T^{2} $
	71	$ 1 + 8.99T + 71T^{2} $
	73	$ 1 + 6.05iT - 73T^{2} $
	79	$ 1 - 8.05T + 79T^{2} $
	83	$ 1 - 1.54iT - 83T^{2} $
	89	$ 1 + 3T + 89T^{2} $
	97	$ 1 + 12.2iT - 97T^{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

−9.158172590744552073952454971832, −8.544398437772833785283059570446, −7.50410582321430165956501677868, −6.43673461627934797391463195625, −5.43304709108912690184173886897, −4.27359861863933360339611885405, −3.88696446125049076306130099017, −2.86914380015348300127184565548, −1.82185258201329858170802531121, −0.924502219627664607472436328785, 0.836334650773962602188042853579, 2.75768526193843834625034241955, 4.09746695747281355116068796809, 4.73794296328958174510432560310, 5.66580231734490741542660672322, 6.33499610360177710016095697608, 6.95891888484664195799996282541, 7.76894590692546002775552160029, 8.385426339423699618621052992204, 9.234066056945084472856173322576

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.594630561\)
\(L(\frac12)\)	\(\approx\)	\(1.594630561\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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