LMFDB - L-function 2-28e2-28.27-c3-0-1

Dirichlet series

L(s) = 1

− 5.68·3^-s + 20.4i·5^-s + 5.37·9^-s − 27.4i·11^-s + 33.2i·13^-s − 116. i·15^-s + 51.6i·17^-s − 123.·19^-s + 203. i·23^-s − 291.·25^-s + 123.·27^-s − 27.1·29^-s − 132.·31^-s + 155. i·33^-s − 98.1·37^-s + ⋯

L(s) = 1

− 1.09·3^-s + 1.82i·5^-s + 0.198·9^-s − 0.751i·11^-s + 0.709i·13^-s − 1.99i·15^-s + 0.736i·17^-s − 1.48·19^-s + 1.84i·23^-s − 2.33·25^-s + 0.877·27^-s − 0.173·29^-s − 0.768·31^-s + 0.822i·33^-s − 0.435·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$784$ = $2^{4} \cdot 7^{2}$
Sign:	$-0.101 + 0.994i$
Analytic conductor:	$46.2574$
Root analytic conductor:	$6.80128$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{784} (783, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 784,\ (\ :3/2),\ -0.101 + 0.994i)$

Particular Values

$L(2)$	$\approx$	$0.1582127385$
$L(\frac12)$	$\approx$	$0.1582127385$
$L(\frac{5}{2})$		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 $
	7	$ 1 $
good	3	$ 1 + 5.68T + 27T^{2} $
	5	$ 1 - 20.4iT - 125T^{2} $
	11	$ 1 + 27.4iT - 1.33e3T^{2} $
	13	$ 1 - 33.2iT - 2.19e3T^{2} $
	17	$ 1 - 51.6iT - 4.91e3T^{2} $
	19	$ 1 + 123.T + 6.85e3T^{2} $
	23	$ 1 - 203. iT - 1.21e4T^{2} $
	29	$ 1 + 27.1T + 2.43e4T^{2} $
	31	$ 1 + 132.T + 2.97e4T^{2} $
	37	$ 1 + 98.1T + 5.06e4T^{2} $
	41	$ 1 - 475. iT - 6.89e4T^{2} $
	43	$ 1 + 288. iT - 7.95e4T^{2} $
	47	$ 1 - 568.T + 1.03e5T^{2} $
	53	$ 1 + 133.T + 1.48e5T^{2} $
	59	$ 1 + 307.T + 2.05e5T^{2} $
	61	$ 1 + 735. iT - 2.26e5T^{2} $
	67	$ 1 - 130. iT - 3.00e5T^{2} $
	71	$ 1 - 708. iT - 3.57e5T^{2} $
	73	$ 1 - 562. iT - 3.89e5T^{2} $
	79	$ 1 + 787. iT - 4.93e5T^{2} $
	83	$ 1 - 300.T + 5.71e5T^{2} $
	89	$ 1 + 1.13e3iT - 7.04e5T^{2} $
	97	$ 1 - 234. iT - 9.12e5T^{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

−10.76165353692718526702317526378, −10.03880956327680359319615468095, −8.875338410682369746798672105318, −7.70835850822912993684336805376, −6.80020545159242180673078759181, −6.20227276162328162204319419530, −5.56922883887660747826819747122, −4.06965050420064758834769430389, −3.15193188900800075225609702355, −1.86726480058289960969021136279, 0.06614178244774067142623802056, 0.814484052640634581904901204186, 2.23315073390137595753118720183, 4.19681104138958063105627102473, 4.84647575602590532601927768876, 5.53513950649831457570047190766, 6.40720817238878284513222207179, 7.58512122089989794126972889290, 8.631984339742916230385135500041, 9.094498447413507005651063032468

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.1582127385\)
\(L(\frac12)\)	\(\approx\)	\(0.1582127385\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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