LMFDB - L-function 2-2100-105.104-c1-0-7

Dirichlet series

L(s) = 1

− 1.73·3^-s + (−2.59 + 0.5i)7^-s + 2.99·9^-s − 1.73·13^-s − 5.19i·19^-s + (4.5 − 0.866i)21^-s − 5.19·27^-s + 1.73i·31^-s + 10i·37^-s + 2.99·39^-s − 13i·43^-s + (6.5 − 2.59i)49^-s + 9i·57^-s + 15.5i·61^-s + (−7.79 + 1.49i)63^-s + ⋯

L(s) = 1

− 1.00·3^-s + (−0.981 + 0.188i)7^-s + 0.999·9^-s − 0.480·13^-s − 1.19i·19^-s + (0.981 − 0.188i)21^-s − 1.00·27^-s + 0.311i·31^-s + 1.64i·37^-s + 0.480·39^-s − 1.98i·43^-s + (0.928 − 0.371i)49^-s + 1.19i·57^-s + 1.99i·61^-s + (−0.981 + 0.188i)63^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2100$ = $2^{2} \cdot 3 \cdot 5^{2} \cdot 7$
Sign:	$0.793 - 0.608i$
Analytic conductor:	$16.7685$
Root analytic conductor:	$4.09494$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2100} (1049, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2100,\ (\ :1/2),\ 0.793 - 0.608i)$

Particular Values

$L(1)$	$\approx$	$0.8177571906$
$L(\frac12)$	$\approx$	$0.8177571906$
$L(\frac{3}{2})$		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 $
	3	$ 1 + 1.73T $
	5	$ 1 $
	7	$ 1 + (2.59 - 0.5i)T $
good	11	$ 1 - 11T^{2} $
	13	$ 1 + 1.73T + 13T^{2} $
	17	$ 1 - 17T^{2} $
	19	$ 1 + 5.19iT - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 - 1.73iT - 31T^{2} $
	37	$ 1 - 10iT - 37T^{2} $
	41	$ 1 + 41T^{2} $
	43	$ 1 + 13iT - 43T^{2} $
	47	$ 1 - 47T^{2} $
	53	$ 1 + 53T^{2} $
	59	$ 1 + 59T^{2} $
	61	$ 1 - 15.5iT - 61T^{2} $
	67	$ 1 - 11iT - 67T^{2} $
	71	$ 1 - 71T^{2} $
	73	$ 1 - 13.8T + 73T^{2} $
	79	$ 1 - 4T + 79T^{2} $
	83	$ 1 - 83T^{2} $
	89	$ 1 + 89T^{2} $
	97	$ 1 - 5.19T + 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

−9.307264854339710089436959992904, −8.527100896738681819499057478511, −7.25158283849292232197273195553, −6.86743380987487298143119307062, −6.03837488155012154171248560544, −5.25711843532051393387026720537, −4.47763042959006819961034600339, −3.41222900930171949937893450307, −2.31236427211878930402396597464, −0.75950971612582345484740653022, 0.49777517793442728489038188339, 1.93743706417033022737543260056, 3.32085605894802316738519550674, 4.16644349459473959673749951746, 5.11153502246802051940146339025, 5.99305260347888915391722498680, 6.48743010089410608242476746622, 7.37830220436601497420856256262, 8.049370582097283859667061288956, 9.428079727605497206573510683283

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8177571906\)
\(L(\frac12)\)	\(\approx\)	\(0.8177571906\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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