LMFDB - L-function 2-1134-21.20-c1-0-5

Dirichlet series

L(s) = 1

− i·2^-s − 4^-s − 1.79·5^-s + (2.28 − 1.34i)7^-s + i·8^-s + 1.79i·10^-s + 2.40i·11^-s + 4.89i·13^-s + (−1.34 − 2.28i)14^-s + 16^-s − 3.66·17^-s − 3.01i·19^-s + 1.79·20^-s + 2.40·22^-s + 3.76i·23^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s − 0.800·5^-s + (0.861 − 0.506i)7^-s + 0.353i·8^-s + 0.566i·10^-s + 0.724i·11^-s + 1.35i·13^-s + (−0.358 − 0.609i)14^-s + 0.250·16^-s − 0.888·17^-s − 0.692i·19^-s + 0.400·20^-s + 0.512·22^-s + 0.785i·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1134$ = $2 \cdot 3^{4} \cdot 7$
Sign:	$0.861 - 0.506i$
Analytic conductor:	$9.05503$
Root analytic conductor:	$3.00915$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1134} (1133, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1134,\ (\ :1/2),\ 0.861 - 0.506i)$

Particular Values

$L(1)$	$\approx$	$1.101296400$
$L(\frac12)$	$\approx$	$1.101296400$
$L(\frac{3}{2})$		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 + iT $
	3	$ 1 $
	7	$ 1 + (-2.28 + 1.34i)T $
good	5	$ 1 + 1.79T + 5T^{2} $
	11	$ 1 - 2.40iT - 11T^{2} $
	13	$ 1 - 4.89iT - 13T^{2} $
	17	$ 1 + 3.66T + 17T^{2} $
	19	$ 1 + 3.01iT - 19T^{2} $
	23	$ 1 - 3.76iT - 23T^{2} $
	29	$ 1 - 6.56iT - 29T^{2} $
	31	$ 1 - 4.64iT - 31T^{2} $
	37	$ 1 - 9.36T + 37T^{2} $
	41	$ 1 + 8.08T + 41T^{2} $
	43	$ 1 - 6.96T + 43T^{2} $
	47	$ 1 - 5.13T + 47T^{2} $
	53	$ 1 - 53T^{2} $
	59	$ 1 - 14.5T + 59T^{2} $
	61	$ 1 - 11.3iT - 61T^{2} $
	67	$ 1 - 0.570T + 67T^{2} $
	71	$ 1 - 5.96iT - 71T^{2} $
	73	$ 1 - 12.3iT - 73T^{2} $
	79	$ 1 - 3.03T + 79T^{2} $
	83	$ 1 + 14.0T + 83T^{2} $
	89	$ 1 - 3.74T + 89T^{2} $
	97	$ 1 - 5.51iT - 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

−9.962214419562297400441608196154, −9.064172918572680437524172945975, −8.421649231649003695210949385895, −7.34453781312353064711171403749, −6.87825224159721582314250999627, −5.28190643704071286697948150977, −4.36392994503983161792084207965, −3.95202382473863069819665520428, −2.44587862063716822258048139052, −1.34023539572199820975066623009, 0.52745538572833701771154465593, 2.44277573605424480342918516880, 3.76874331127379967702387697911, 4.60637174410768822904483037425, 5.63484384851974207858251660463, 6.24178992835011105998742470677, 7.55318511959564346381203427047, 8.092468521843159191880226313094, 8.522377558899195872442923222867, 9.595498349684490736095710357107

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.101296400\)
\(L(\frac12)\)	\(\approx\)	\(1.101296400\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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