LMFDB - L-function 2-4704-8.5-c1-0-56

Dirichlet series

L(s) = 1

− i·3^-s − 3.56i·5^-s − 9^-s + 4.07i·11^-s + 1.44i·13^-s − 3.56·15^-s + 6.99·17^-s − 0.301i·19^-s + 2.42·23^-s − 7.71·25^-s + i·27^-s − 0.151i·29^-s + 4.75·31^-s + 4.07·33^-s − 11.3i·37^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 1.59i·5^-s − 0.333·9^-s + 1.22i·11^-s + 0.399i·13^-s − 0.920·15^-s + 1.69·17^-s − 0.0691i·19^-s + 0.505·23^-s − 1.54·25^-s + 0.192i·27^-s − 0.0281i·29^-s + 0.853·31^-s + 0.708·33^-s − 1.86i·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4704$ = $2^{5} \cdot 3 \cdot 7^{2}$
Sign:	$-0.167 + 0.985i$
Analytic conductor:	$37.5616$
Root analytic conductor:	$6.12875$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4704} (2353, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4704,\ (\ :1/2),\ -0.167 + 0.985i)$

Particular Values

$L(1)$	$\approx$	$1.985792024$
$L(\frac12)$	$\approx$	$1.985792024$
$L(\frac{3}{2})$		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 + iT $
	7	$ 1 $
good	5	$ 1 + 3.56iT - 5T^{2} $
	11	$ 1 - 4.07iT - 11T^{2} $
	13	$ 1 - 1.44iT - 13T^{2} $
	17	$ 1 - 6.99T + 17T^{2} $
	19	$ 1 + 0.301iT - 19T^{2} $
	23	$ 1 - 2.42T + 23T^{2} $
	29	$ 1 + 0.151iT - 29T^{2} $
	31	$ 1 - 4.75T + 31T^{2} $
	37	$ 1 + 11.3iT - 37T^{2} $
	41	$ 1 + 0.239T + 41T^{2} $
	43	$ 1 - 1.32iT - 43T^{2} $
	47	$ 1 - 6.35T + 47T^{2} $
	53	$ 1 + 5.98iT - 53T^{2} $
	59	$ 1 + 11.2iT - 59T^{2} $
	61	$ 1 - 4.20iT - 61T^{2} $
	67	$ 1 - 4.38iT - 67T^{2} $
	71	$ 1 - 8.46T + 71T^{2} $
	73	$ 1 - 0.569T + 73T^{2} $
	79	$ 1 - 1.49T + 79T^{2} $
	83	$ 1 - 10.0iT - 83T^{2} $
	89	$ 1 + 3.66T + 89T^{2} $
	97	$ 1 - 10.4T + 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

−8.057781249633178502164718791309, −7.49693753625722473610621008144, −6.77221698894041718347331819934, −5.72723730373669615308230532822, −5.19568661668634670267914024759, −4.48950758038903653648438604569, −3.66272128370356788873104410856, −2.35963088742215461343789590482, −1.48976265907086306670086028925, −0.68027434446858472135392623608, 1.01040479741189119669903487241, 2.63166986880982531169033347498, 3.19120709086259486556519481486, 3.64540869583253469488835911102, 4.84191507475842540682013939435, 5.76974268370130023127632393282, 6.18550633389185939153615370386, 7.04893094832467609063033300029, 7.81140547493972191275407285677, 8.372627128394825407324233998140

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.985792024\)
\(L(\frac12)\)	\(\approx\)	\(1.985792024\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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