LMFDB - L-function 2-1850-185.184-c1-0-26

Dirichlet series

L(s) = 1

− 2^-s − 1.79i·3^-s + 4^-s + 1.79i·6^-s − 2i·7^-s − 8^-s − 0.208·9^-s − 0.791·11^-s − 1.79i·12^-s + 3.79·13^-s + 2i·14^-s + 16^-s + 7.58·17^-s + 0.208·18^-s + 1.58i·19^-s + ⋯

L(s) = 1

− 0.707·2^-s − 1.03i·3^-s + 0.5·4^-s + 0.731i·6^-s − 0.755i·7^-s − 0.353·8^-s − 0.0695·9^-s − 0.238·11^-s − 0.517i·12^-s + 1.05·13^-s + 0.534i·14^-s + 0.250·16^-s + 1.83·17^-s + 0.0491·18^-s + 0.363i·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1850$ = $2 \cdot 5^{2} \cdot 37$
Sign:	$0.251 + 0.967i$
Analytic conductor:	$14.7723$
Root analytic conductor:	$3.84347$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1850} (1849, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1850,\ (\ :1/2),\ 0.251 + 0.967i)$

Particular Values

$L(1)$	$\approx$	$1.478239789$
$L(\frac12)$	$\approx$	$1.478239789$
$L(\frac{3}{2})$		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 + T $
	5	$ 1 $
	37	$ 1 + (4.58 - 4i)T $
good	3	$ 1 + 1.79iT - 3T^{2} $
	7	$ 1 + 2iT - 7T^{2} $
	11	$ 1 + 0.791T + 11T^{2} $
	13	$ 1 - 3.79T + 13T^{2} $
	17	$ 1 - 7.58T + 17T^{2} $
	19	$ 1 - 1.58iT - 19T^{2} $
	23	$ 1 - 3.79T + 23T^{2} $
	29	$ 1 - 3.79iT - 29T^{2} $
	31	$ 1 - 8.37iT - 31T^{2} $
	41	$ 1 - 9.79T + 41T^{2} $
	43	$ 1 - 6T + 43T^{2} $
	47	$ 1 + 7.58iT - 47T^{2} $
	53	$ 1 - 1.58iT - 53T^{2} $
	59	$ 1 - 1.58iT - 59T^{2} $
	61	$ 1 + 12.7iT - 61T^{2} $
	67	$ 1 - 6.37iT - 67T^{2} $
	71	$ 1 + 9.16T + 71T^{2} $
	73	$ 1 - 4.37iT - 73T^{2} $
	79	$ 1 - 8.20iT - 79T^{2} $
	83	$ 1 + 15.1iT - 83T^{2} $
	89	$ 1 - 6iT - 89T^{2} $
	97	$ 1 + 13.5T + 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

−8.888666966716064462674387534669, −8.191168453859459378550413625021, −7.45497288386725234758452110104, −7.01751538408779200946231560740, −6.14191413084126799398457319412, −5.26056707444456401867074770452, −3.85152413883544788842529974530, −2.95435463888031708708316657493, −1.48033407367931156603513541104, −0.965186453029696974935090366476, 1.08006732568262973757802074779, 2.54640881023100922478617978537, 3.49649346856778888315106579426, 4.40241692955685151643050825147, 5.59491399443161098657031618404, 5.95691004138837946483932858157, 7.30951677916609647889991115261, 7.941510042612963320043059825940, 8.921859067905269832160794505280, 9.351612343655988270590881805758

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.478239789\)
\(L(\frac12)\)	\(\approx\)	\(1.478239789\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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