LMFDB - L-function 2-825-5.4-c3-0-19

Dirichlet series

L(s) = 1

− 4.38i·2^-s + 3i·3^-s − 11.2·4^-s + 13.1·6^-s − 11.7i·7^-s + 14.2i·8^-s − 9·9^-s + 11·11^-s − 33.7i·12^-s + 72.8i·13^-s − 51.4·14^-s − 27.3·16^-s − 9.89i·17^-s + 39.4i·18^-s − 0.0238·19^-s + ⋯

L(s) = 1

− 1.55i·2^-s + 0.577i·3^-s − 1.40·4^-s + 0.895·6^-s − 0.633i·7^-s + 0.631i·8^-s − 0.333·9^-s + 0.301·11^-s − 0.812i·12^-s + 1.55i·13^-s − 0.982·14^-s − 0.426·16^-s − 0.141i·17^-s + 0.517i·18^-s − 0.000288·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$825$ = $3 \cdot 5^{2} \cdot 11$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$48.6765$
Root analytic conductor:	$6.97686$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{825} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 825,\ (\ :3/2),\ 0.894 + 0.447i)$

Particular Values

$L(2)$	$\approx$	$1.470245304$
$L(\frac12)$	$\approx$	$1.470245304$
$L(\frac{5}{2})$		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 - 3iT $
	5	$ 1 $
	11	$ 1 - 11T $
good	2	$ 1 + 4.38iT - 8T^{2} $
	7	$ 1 + 11.7iT - 343T^{2} $
	13	$ 1 - 72.8iT - 2.19e3T^{2} $
	17	$ 1 + 9.89iT - 4.91e3T^{2} $
	19	$ 1 + 0.0238T + 6.85e3T^{2} $
	23	$ 1 + 73.0iT - 1.21e4T^{2} $
	29	$ 1 + 202.T + 2.43e4T^{2} $
	31	$ 1 - 181.T + 2.97e4T^{2} $
	37	$ 1 - 299. iT - 5.06e4T^{2} $
	41	$ 1 - 88.5T + 6.89e4T^{2} $
	43	$ 1 - 146. iT - 7.95e4T^{2} $
	47	$ 1 - 185. iT - 1.03e5T^{2} $
	53	$ 1 - 347. iT - 1.48e5T^{2} $
	59	$ 1 + 691.T + 2.05e5T^{2} $
	61	$ 1 - 491.T + 2.26e5T^{2} $
	67	$ 1 - 715. iT - 3.00e5T^{2} $
	71	$ 1 - 541.T + 3.57e5T^{2} $
	73	$ 1 - 159. iT - 3.89e5T^{2} $
	79	$ 1 - 212.T + 4.93e5T^{2} $
	83	$ 1 + 413. iT - 5.71e5T^{2} $
	89	$ 1 - 1.09e3T + 7.04e5T^{2} $
	97	$ 1 - 567. iT - 9.12e5T^{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

−9.878796884349014903338724412477, −9.342139038748520502213393055616, −8.538202627054264409292146970457, −7.20037113926887835128432236946, −6.23387546928713602285425681578, −4.64799749478372910708621657688, −4.21859974898862005948027952461, −3.24578303798602219796874319867, −2.12017290938942708989069213662, −0.984575973238767577934621461867, 0.46888439050595172039664651495, 2.20446374718545994646744891923, 3.59569481581136268637450347590, 5.09322332178510061089679187157, 5.70191986036564113213159354284, 6.38474898242672282771120981820, 7.42177892801253557396638038632, 7.910504469344984720289546022607, 8.732027776229399882564911168074, 9.476800366853235486167951393374

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

\(L(2)\)	\(\approx\)	\(1.470245304\)
\(L(\frac12)\)	\(\approx\)	\(1.470245304\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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