LMFDB - L-function 2-3800-5.4-c1-0-17

Dirichlet series

L(s) = 1

− 3.29i·3^-s + 1.78i·7^-s − 7.87·9^-s − 5.08·11^-s + 1.29i·13^-s + 0.213i·17^-s + 19^-s + 5.89·21^-s + 3.72i·23^-s + 16.0i·27^-s − 0.870·29^-s + 16.7i·33^-s − 2i·37^-s + 4.27·39^-s + 8.59·41^-s + ⋯

L(s) = 1

− 1.90i·3^-s + 0.675i·7^-s − 2.62·9^-s − 1.53·11^-s + 0.359i·13^-s + 0.0517i·17^-s + 0.229·19^-s + 1.28·21^-s + 0.776i·23^-s + 3.09i·27^-s − 0.161·29^-s + 2.91i·33^-s − 0.328i·37^-s + 0.684·39^-s + 1.34·41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3800$ = $2^{3} \cdot 5^{2} \cdot 19$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$30.3431$
Root analytic conductor:	$5.50846$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3800} (3649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3800,\ (\ :1/2),\ 0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$1.142898191$
$L(\frac12)$	$\approx$	$1.142898191$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	5	$ 1 $
	19	$ 1 - T $
good	3	$ 1 + 3.29iT - 3T^{2} $
	7	$ 1 - 1.78iT - 7T^{2} $
	11	$ 1 + 5.08T + 11T^{2} $
	13	$ 1 - 1.29iT - 13T^{2} $
	17	$ 1 - 0.213iT - 17T^{2} $
	23	$ 1 - 3.72iT - 23T^{2} $
	29	$ 1 + 0.870T + 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 2iT - 37T^{2} $
	41	$ 1 - 8.59T + 41T^{2} $
	43	$ 1 - 3.67iT - 43T^{2} $
	47	$ 1 + 4.65iT - 47T^{2} $
	53	$ 1 + 11.0iT - 53T^{2} $
	59	$ 1 - 4.70T + 59T^{2} $
	61	$ 1 - 3.51T + 61T^{2} $
	67	$ 1 - 1.12iT - 67T^{2} $
	71	$ 1 - 8.76T + 71T^{2} $
	73	$ 1 + 6.80iT - 73T^{2} $
	79	$ 1 + 14.5T + 79T^{2} $
	83	$ 1 - 9.74iT - 83T^{2} $
	89	$ 1 - 6.76T + 89T^{2} $
	97	$ 1 - 4.16iT - 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

−8.242831992292813961791070088434, −7.62961825898824289645098686927, −7.14400245054850604366597799711, −6.25877460940886423416976043402, −5.62708923985748856482085766401, −5.05764035846719257121564050303, −3.46891268047027817947593825152, −2.47746876956439835558153654184, −2.06849824965731914520627372508, −0.813819482521378409310606198258, 0.43633270999248396691422672889, 2.54285430051668785870194570838, 3.14327578219914648328075433019, 4.12452443222169320759764751278, 4.62215930456886154674755318837, 5.42774559470322649013999145821, 5.93837314938285479795521522476, 7.23023390637691402810923139053, 8.011016614926993063084647821514, 8.645238315829141040651353748430

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.142898191\)
\(L(\frac12)\)	\(\approx\)	\(1.142898191\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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