LMFDB - L-function 2-2e8-8.5-c7-0-24

Dirichlet series

L(s) = 1

− 62.2i·3^-s − 203. i·5^-s + 534.·7^-s − 1.68e3·9^-s + 1.57e3i·11^-s + 1.77e3i·13^-s − 1.26e4·15^-s + 2.89e4·17^-s + 3.92e4i·19^-s − 3.32e4i·21^-s + 8.52e4·23^-s + 3.69e4·25^-s − 3.09e4i·27^-s + 2.46e5i·29^-s + 5.46e4·31^-s + ⋯

L(s) = 1

− 1.33i·3^-s − 0.726i·5^-s + 0.589·7^-s − 0.772·9^-s + 0.357i·11^-s + 0.223i·13^-s − 0.966·15^-s + 1.43·17^-s + 1.31i·19^-s − 0.784i·21^-s + 1.46·23^-s + 0.472·25^-s − 0.303i·27^-s + 1.87i·29^-s + 0.329·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$256$ = $2^{8}$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$79.9705$
Root analytic conductor:	$8.94262$
Motivic weight:	$7$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{256} (129, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 256,\ (\ :7/2),\ 0.707 + 0.707i)$

Particular Values

$L(4)$	$\approx$	$2.619577539$
$L(\frac12)$	$\approx$	$2.619577539$
$L(\frac{9}{2})$		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 $
good	3	$ 1 + 62.2iT - 2.18e3T^{2} $
	5	$ 1 + 203. iT - 7.81e4T^{2} $
	7	$ 1 - 534.T + 8.23e5T^{2} $
	11	$ 1 - 1.57e3iT - 1.94e7T^{2} $
	13	$ 1 - 1.77e3iT - 6.27e7T^{2} $
	17	$ 1 - 2.89e4T + 4.10e8T^{2} $
	19	$ 1 - 3.92e4iT - 8.93e8T^{2} $
	23	$ 1 - 8.52e4T + 3.40e9T^{2} $
	29	$ 1 - 2.46e5iT - 1.72e10T^{2} $
	31	$ 1 - 5.46e4T + 2.75e10T^{2} $
	37	$ 1 - 2.90e5iT - 9.49e10T^{2} $
	41	$ 1 - 7.25e5T + 1.94e11T^{2} $
	43	$ 1 - 6.68e5iT - 2.71e11T^{2} $
	47	$ 1 + 4.29e5T + 5.06e11T^{2} $
	53	$ 1 - 7.25e5iT - 1.17e12T^{2} $
	59	$ 1 - 1.65e5iT - 2.48e12T^{2} $
	61	$ 1 - 2.66e6iT - 3.14e12T^{2} $
	67	$ 1 + 1.16e6iT - 6.06e12T^{2} $
	71	$ 1 - 3.87e6T + 9.09e12T^{2} $
	73	$ 1 + 5.79e5T + 1.10e13T^{2} $
	79	$ 1 + 1.92e6T + 1.92e13T^{2} $
	83	$ 1 + 4.86e6iT - 2.71e13T^{2} $
	89	$ 1 + 4.58e6T + 4.42e13T^{2} $
	97	$ 1 + 4.18e6T + 8.07e13T^{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

−10.80367760955281309347884828781, −9.559523454494046794510419383539, −8.425162619692323596123237835329, −7.73005869623574463165042967939, −6.85116287923830047890835858442, −5.65203757594413876791201718042, −4.64618018801703275784393023451, −3.00159384701357200939345191040, −1.36891822380110051748791910257, −1.18734159288340735182814383141, 0.72956890728057324069030438090, 2.63418767984910444257120306746, 3.58385466286082233194605839874, 4.71838532335857996626725777808, 5.57622454643855621196580708641, 6.95656761411021750135330517477, 8.068709081733376495561147328475, 9.218120877931517428057161772983, 9.969238398440249757532582160607, 10.97125775257670835952716741243

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(2.619577539\)
\(L(\frac12)\)	\(\approx\)	\(2.619577539\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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