LMFDB - L-function 2-1200-12.11-c1-0-25

Dirichlet series

L(s) = 1

+ 1.73i·3^-s − 3.46i·7^-s − 2.99·9^-s + 2·13^-s − 3.46i·19^-s + 5.99·21^-s − 5.19i·27^-s − 10.3i·31^-s + 10·37^-s + 3.46i·39^-s − 10.3i·43^-s − 4.99·49^-s + 5.99·57^-s + 14·61^-s + 10.3i·63^-s + ⋯

L(s) = 1

+ 0.999i·3^-s − 1.30i·7^-s − 0.999·9^-s + 0.554·13^-s − 0.794i·19^-s + 1.30·21^-s − 0.999i·27^-s − 1.86i·31^-s + 1.64·37^-s + 0.554i·39^-s − 1.58i·43^-s − 0.714·49^-s + 0.794·57^-s + 1.79·61^-s + 1.30i·63^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1200$ = $2^{4} \cdot 3 \cdot 5^{2}$
Sign:	$0.866 + 0.5i$
Analytic conductor:	$9.58204$
Root analytic conductor:	$3.09548$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1200} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1200,\ (\ :1/2),\ 0.866 + 0.5i)$

Particular Values

$L(1)$	$\approx$	$1.429421661$
$L(\frac12)$	$\approx$	$1.429421661$
$L(\frac{3}{2})$		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 - 1.73iT $
	5	$ 1 $
good	7	$ 1 + 3.46iT - 7T^{2} $
	11	$ 1 + 11T^{2} $
	13	$ 1 - 2T + 13T^{2} $
	17	$ 1 - 17T^{2} $
	19	$ 1 + 3.46iT - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 + 10.3iT - 31T^{2} $
	37	$ 1 - 10T + 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 + 10.3iT - 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 53T^{2} $
	59	$ 1 + 59T^{2} $
	61	$ 1 - 14T + 61T^{2} $
	67	$ 1 - 3.46iT - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 10T + 73T^{2} $
	79	$ 1 - 17.3iT - 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 - 89T^{2} $
	97	$ 1 - 14T + 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

−9.806252096077230051549615030773, −9.004961524472074192758532613896, −8.109096388454995116136049978616, −7.26195777582282358024226536639, −6.27345544268238272974020158098, −5.32281012845077203772794084647, −4.23829177968066935399154452744, −3.83186428156488392870057782850, −2.55365850711334460748557892017, −0.66784532134024500454975933619, 1.33600478664384813188412092758, 2.43542790409807925484042560071, 3.36105338282589031889763531663, 4.88789315724740340779684229735, 5.87738937373433097079196097912, 6.31757887149443789052477207507, 7.39194679382306185360896595271, 8.278234613837785937109730992817, 8.762550242819512383214775020218, 9.648419446908427041658540004991

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.429421661\)
\(L(\frac12)\)	\(\approx\)	\(1.429421661\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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