LMFDB - L-function 2-3800-5.4-c1-0-62

Dirichlet series

L(s) = 1

+ 3.08i·3^-s + 4.29i·7^-s − 6.51·9^-s − 1.21·11^-s − 5.08i·13^-s − 2.29i·17^-s + 19^-s − 13.2·21^-s − 7.67i·23^-s − 10.8i·27^-s + 0.489·29^-s − 3.74i·33^-s − 2i·37^-s + 15.6·39^-s − 4.16·41^-s + ⋯

L(s) = 1

+ 1.78i·3^-s + 1.62i·7^-s − 2.17·9^-s − 0.365·11^-s − 1.41i·13^-s − 0.557i·17^-s + 0.229·19^-s − 2.89·21^-s − 1.60i·23^-s − 2.08i·27^-s + 0.0909·29^-s − 0.651i·33^-s − 0.328i·37^-s + 2.51·39^-s − 0.650·41^-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

$L(1)$	$\approx$	$0.6123491332$
$L(\frac12)$	$\approx$	$0.6123491332$
$L(\frac{3}{2})$		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 - 3.08iT - 3T^{2} $
	7	$ 1 - 4.29iT - 7T^{2} $
	11	$ 1 + 1.21T + 11T^{2} $
	13	$ 1 + 5.08iT - 13T^{2} $
	17	$ 1 + 2.29iT - 17T^{2} $
	23	$ 1 + 7.67iT - 23T^{2} $
	29	$ 1 - 0.489T + 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 2iT - 37T^{2} $
	41	$ 1 + 4.16T + 41T^{2} $
	43	$ 1 + 12.9iT - 43T^{2} $
	47	$ 1 + 5.80iT - 47T^{2} $
	53	$ 1 + 1.93iT - 53T^{2} $
	59	$ 1 - 11.0T + 59T^{2} $
	61	$ 1 + 5.38T + 61T^{2} $
	67	$ 1 - 2.48iT - 67T^{2} $
	71	$ 1 + 11.7T + 71T^{2} $
	73	$ 1 - 8.46iT - 73T^{2} $
	79	$ 1 + 1.83T + 79T^{2} $
	83	$ 1 - 7.02iT - 83T^{2} $
	89	$ 1 + 13.7T + 89T^{2} $
	97	$ 1 + 3.57iT - 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

−8.590870865685355767489217060915, −8.179836837340462387887825771707, −6.86659583918618434549370964103, −5.61204123343592595059291395406, −5.52510651675324663879447766850, −4.79207056184273708729411307530, −3.81834009142702967632493230365, −2.89920171658089562192851956218, −2.45630065209201146777977592173, −0.18117613672399972289656012796, 1.23088984517142501884850294330, 1.64346976005425093249839407508, 2.89891829240674223251911496352, 3.85676706846165223567941759290, 4.76911716785583111884395203470, 5.91339900931436587634682368140, 6.56008320259746766802898049846, 7.16627588304168852570438866281, 7.63160751573488777814412398505, 8.189768577543337456022691851440

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6123491332\)
\(L(\frac12)\)	\(\approx\)	\(0.6123491332\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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