LMFDB - L-function 2-3800-5.4-c1-0-60

Dirichlet series

L(s) = 1

− 0.848i·3^-s + 1.74i·7^-s + 2.28·9^-s + 5.92·11^-s − 6.78i·13^-s + 1.86i·17^-s + 19^-s + 1.48·21^-s − 5.94i·23^-s − 4.47i·27^-s − 3.29·29^-s − 5.75·31^-s − 5.02i·33^-s − 4.36i·37^-s − 5.75·39^-s + ⋯

L(s) = 1

− 0.489i·3^-s + 0.659i·7^-s + 0.760·9^-s + 1.78·11^-s − 1.88i·13^-s + 0.451i·17^-s + 0.229·19^-s + 0.322·21^-s − 1.23i·23^-s − 0.862i·27^-s − 0.612·29^-s − 1.03·31^-s − 0.874i·33^-s − 0.717i·37^-s − 0.921·39^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$3800$ = $2^{3} \cdot 5^{2} \cdot 19$
Sign:	$0.447 + 0.894i$
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:	$(2,\ 3800,\ (\ :1/2),\ 0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$2.270915742$
$L(\frac12)$	$\approx$	$2.270915742$
$L(\frac{3}{2})$		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 + 0.848iT - 3T^{2} $
	7	$ 1 - 1.74iT - 7T^{2} $
	11	$ 1 - 5.92T + 11T^{2} $
	13	$ 1 + 6.78iT - 13T^{2} $
	17	$ 1 - 1.86iT - 17T^{2} $
	23	$ 1 + 5.94iT - 23T^{2} $
	29	$ 1 + 3.29T + 29T^{2} $
	31	$ 1 + 5.75T + 31T^{2} $
	37	$ 1 + 4.36iT - 37T^{2} $
	41	$ 1 - 7.12T + 41T^{2} $
	43	$ 1 - 6.98iT - 43T^{2} $
	47	$ 1 + 4.02iT - 47T^{2} $
	53	$ 1 - 9.19iT - 53T^{2} $
	59	$ 1 + 2.51T + 59T^{2} $
	61	$ 1 + 2.49T + 61T^{2} $
	67	$ 1 - 6.90iT - 67T^{2} $
	71	$ 1 - 1.27T + 71T^{2} $
	73	$ 1 + 12.1iT - 73T^{2} $
	79	$ 1 - 13.8T + 79T^{2} $
	83	$ 1 - 4.94iT - 83T^{2} $
	89	$ 1 + 15.6T + 89T^{2} $
	97	$ 1 + 15.7iT - 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

−8.324655027614379535576030737199, −7.60869879361256874068967772749, −6.93046094990278175217397234404, −6.08716290866800015680594243099, −5.63440434947250956652456405853, −4.46613191150594795364263193140, −3.74390137480486905519075226980, −2.74812043531637632592192925961, −1.69903707499535586907531091451, −0.75878320751210343692317020396, 1.22105117527398968286102999411, 1.91303258702036406006892527806, 3.60018044465088418457920987096, 3.96796751526747506118423040006, 4.58887128385372017041781616548, 5.60095463038775083830483832420, 6.80304683040596534375394544574, 6.87712206170750253262522843812, 7.74511386516323201435774703777, 9.014812229529090606949252830806

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.270915742\)
\(L(\frac12)\)	\(\approx\)	\(2.270915742\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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