LMFDB - L-function 2-2300-115.114-c2-0-13

Dirichlet series

L(s) = 1

+ 1.54i·3^-s − 7.17·7^-s + 6.62·9^-s + 14.0i·11^-s + 20.3i·13^-s − 20.3·17^-s + 0.231i·19^-s − 11.0i·21^-s + (22.0 + 6.58i)23^-s + 24.0i·27^-s + 40.6·29^-s − 0.922·31^-s − 21.6·33^-s − 7.73·37^-s − 31.4·39^-s + ⋯

L(s) = 1

+ 0.514i·3^-s − 1.02·7^-s + 0.735·9^-s + 1.27i·11^-s + 1.56i·13^-s − 1.19·17^-s + 0.0122i·19^-s − 0.526i·21^-s + (0.958 + 0.286i)23^-s + 0.892i·27^-s + 1.40·29^-s − 0.0297·31^-s − 0.656·33^-s − 0.209·37^-s − 0.806·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2300$ = $2^{2} \cdot 5^{2} \cdot 23$
Sign:	$-0.985 + 0.172i$
Analytic conductor:	$62.6704$
Root analytic conductor:	$7.91646$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2300} (1149, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2300,\ (\ :1),\ -0.985 + 0.172i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.9703269769$
$L(\frac12)$	$\approx$	$0.9703269769$
$L(2)$		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 $
	23	$ 1 + (-22.0 - 6.58i)T $
good	3	$ 1 - 1.54iT - 9T^{2} $
	7	$ 1 + 7.17T + 49T^{2} $
	11	$ 1 - 14.0iT - 121T^{2} $
	13	$ 1 - 20.3iT - 169T^{2} $
	17	$ 1 + 20.3T + 289T^{2} $
	19	$ 1 - 0.231iT - 361T^{2} $
	29	$ 1 - 40.6T + 841T^{2} $
	31	$ 1 + 0.922T + 961T^{2} $
	37	$ 1 + 7.73T + 1.36e3T^{2} $
	41	$ 1 + 1.20T + 1.68e3T^{2} $
	43	$ 1 - 17.6T + 1.84e3T^{2} $
	47	$ 1 + 14.9iT - 2.20e3T^{2} $
	53	$ 1 + 58.6T + 2.80e3T^{2} $
	59	$ 1 + 16.7T + 3.48e3T^{2} $
	61	$ 1 + 39.5iT - 3.72e3T^{2} $
	67	$ 1 + 45.2T + 4.48e3T^{2} $
	71	$ 1 - 43.5T + 5.04e3T^{2} $
	73	$ 1 + 60.0iT - 5.32e3T^{2} $
	79	$ 1 - 49.3iT - 6.24e3T^{2} $
	83	$ 1 - 56.4T + 6.88e3T^{2} $
	89	$ 1 - 65.3iT - 7.92e3T^{2} $
	97	$ 1 + 128.T + 9.40e3T^{2} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−9.427520501508092131171935427889, −8.760703465374437250028133023576, −7.49713306346215541059099964705, −6.67756379316118702541058236826, −6.54885569024010270873672220467, −4.94715339733567851580943085550, −4.47736324723688031752134975773, −3.69901660978137609170317913016, −2.52094950980236109248305346133, −1.51291382514066179862970461562, 0.25675536278991641114161651132, 1.11841154959681694748823689335, 2.68029949334242585330318279243, 3.22782030141423384599675797661, 4.34583126406315633816292240203, 5.37740208884625883165937373999, 6.31037695982856698024255684224, 6.69736191114232512349264993357, 7.66211026372859258828053614060, 8.394636872890169412227273056968

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.9703269769\)
\(L(\frac12)\)	\(\approx\)	\(0.9703269769\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more