LMFDB - L-function 2-1152-8.5-c3-0-12

Dirichlet series

L(s) = 1

− 22i·5^-s + 92i·13^-s + 104·17^-s − 359·25^-s + 130i·29^-s + 396i·37^-s − 472·41^-s − 343·49^-s + 518i·53^-s + 468i·61^-s + 2.02e3·65^-s + 1.09e3·73^-s − 2.28e3i·85^-s − 176·89^-s + 594·97^-s + ⋯

L(s) = 1

− 1.96i·5^-s + 1.96i·13^-s + 1.48·17^-s − 2.87·25^-s + 0.832i·29^-s + 1.75i·37^-s − 1.79·41^-s − 49^-s + 1.34i·53^-s + 0.982i·61^-s + 3.86·65^-s + 1.76·73^-s − 2.91i·85^-s − 0.209·89^-s + 0.621·97^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1152$ = $2^{7} \cdot 3^{2}$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$67.9702$
Root analytic conductor:	$8.24440$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1152} (577, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1152,\ (\ :3/2),\ 0.707 - 0.707i)$

Particular Values

$L(2)$	$\approx$	$1.496798537$
$L(\frac12)$	$\approx$	$1.496798537$
$L(\frac{5}{2})$		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 $
	3	$ 1 $
good	5	$ 1 + 22iT - 125T^{2} $
	7	$ 1 + 343T^{2} $
	11	$ 1 - 1.33e3T^{2} $
	13	$ 1 - 92iT - 2.19e3T^{2} $
	17	$ 1 - 104T + 4.91e3T^{2} $
	19	$ 1 - 6.85e3T^{2} $
	23	$ 1 + 1.21e4T^{2} $
	29	$ 1 - 130iT - 2.43e4T^{2} $
	31	$ 1 + 2.97e4T^{2} $
	37	$ 1 - 396iT - 5.06e4T^{2} $
	41	$ 1 + 472T + 6.89e4T^{2} $
	43	$ 1 - 7.95e4T^{2} $
	47	$ 1 + 1.03e5T^{2} $
	53	$ 1 - 518iT - 1.48e5T^{2} $
	59	$ 1 - 2.05e5T^{2} $
	61	$ 1 - 468iT - 2.26e5T^{2} $
	67	$ 1 - 3.00e5T^{2} $
	71	$ 1 + 3.57e5T^{2} $
	73	$ 1 - 1.09e3T + 3.89e5T^{2} $
	79	$ 1 + 4.93e5T^{2} $
	83	$ 1 - 5.71e5T^{2} $
	89	$ 1 + 176T + 7.04e5T^{2} $
	97	$ 1 - 594T + 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

−9.459575567983006801211560838401, −8.694770171897172745619544128182, −8.153446041973150164978631847715, −7.08276952125274223408447709788, −6.03161652659051841864626223642, −5.04436048771782344601069793970, −4.54563533469717558309558860576, −3.50755159231955498694552543596, −1.75864231969248505801397874591, −1.11774815787704034665666049057, 0.38698956432979135456994068457, 2.14246599443990054505614360189, 3.21580376133176961370314166557, 3.55210932264918301656539141648, 5.29994290890271469750969986721, 5.98153055837876300370539954559, 6.86029747532526183396515388647, 7.71420663486240630148559518068, 8.123635703419419196953461610066, 9.732953960746515101223919517471

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.496798537\)
\(L(\frac12)\)	\(\approx\)	\(1.496798537\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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