LMFDB - L-function 2-3800-5.4-c1-0-31

Dirichlet series

L(s) = 1

− 3.26i·3^-s + 4.07i·7^-s − 7.63·9^-s − 0.786·11^-s − 1.07i·13^-s − 1.90i·17^-s + 19^-s + 13.2·21^-s + 1.41i·23^-s + 15.1i·27^-s + 7.26·29^-s + 2.22·31^-s + 2.56i·33^-s + 9.14i·37^-s − 3.52·39^-s + ⋯

L(s) = 1

− 1.88i·3^-s + 1.53i·7^-s − 2.54·9^-s − 0.237·11^-s − 0.299i·13^-s − 0.462i·17^-s + 0.229·19^-s + 2.89·21^-s + 0.294i·23^-s + 2.91i·27^-s + 1.34·29^-s + 0.398·31^-s + 0.446i·33^-s + 1.50i·37^-s − 0.563·39^-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

$L(1)$	$\approx$	$1.614798331$
$L(\frac12)$	$\approx$	$1.614798331$
$L(\frac{3}{2})$		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 + 3.26iT - 3T^{2} $
	7	$ 1 - 4.07iT - 7T^{2} $
	11	$ 1 + 0.786T + 11T^{2} $
	13	$ 1 + 1.07iT - 13T^{2} $
	17	$ 1 + 1.90iT - 17T^{2} $
	23	$ 1 - 1.41iT - 23T^{2} $
	29	$ 1 - 7.26T + 29T^{2} $
	31	$ 1 - 2.22T + 31T^{2} $
	37	$ 1 - 9.14iT - 37T^{2} $
	41	$ 1 + 6.11T + 41T^{2} $
	43	$ 1 + 8.40iT - 43T^{2} $
	47	$ 1 - 3.56iT - 47T^{2} $
	53	$ 1 - 8.57iT - 53T^{2} $
	59	$ 1 - 13.4T + 59T^{2} $
	61	$ 1 - 12.7T + 61T^{2} $
	67	$ 1 + 5.10iT - 67T^{2} $
	71	$ 1 - 1.65T + 71T^{2} $
	73	$ 1 + 10.3iT - 73T^{2} $
	79	$ 1 - 16.2T + 79T^{2} $
	83	$ 1 + 9.35iT - 83T^{2} $
	89	$ 1 + 3.10T + 89T^{2} $
	97	$ 1 - 4.55iT - 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

−8.361759919392963175043582249616, −7.66816361116193800189019729365, −6.81445518026852489047108099304, −6.30910268864707600655478187169, −5.53900090328143575505763718783, −4.98815138538707433035922635445, −3.20657337551944418323733181437, −2.59487011260660598725566595155, −1.87405592802922206409215527093, −0.75438808821020124960839066301, 0.70353902084649560985104790644, 2.47885135773356227832678197441, 3.60768873913419266351160313048, 3.94063397802300854655965455235, 4.73176473210378900860893916074, 5.30792515859795178943891170717, 6.35529882300263165872874034910, 7.11695215715282357068528735790, 8.211160672116515926979060140759, 8.601373850428433226164719954342

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

\(L(1)\)	\(\approx\)	\(1.614798331\)
\(L(\frac12)\)	\(\approx\)	\(1.614798331\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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