LMFDB - L-function 2-1200-5.4-c3-0-11

Dirichlet series

L(s) = 1

− 3i·3^-s + 5i·7^-s − 9·9^-s − 14·11^-s + i·13^-s − 46i·17^-s + 19·19^-s + 15·21^-s + 46i·23^-s + 27i·27^-s − 14·29^-s − 133·31^-s + 42i·33^-s − 258i·37^-s + 3·39^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.269i·7^-s − 0.333·9^-s − 0.383·11^-s + 0.0213i·13^-s − 0.656i·17^-s + 0.229·19^-s + 0.155·21^-s + 0.417i·23^-s + 0.192i·27^-s − 0.0896·29^-s − 0.770·31^-s + 0.221i·33^-s − 1.14i·37^-s + 0.0123·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1200$ = $2^{4} \cdot 3 \cdot 5^{2}$
Sign:	$0.447 - 0.894i$
Analytic conductor:	$70.8022$
Root analytic conductor:	$8.41440$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1200} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1200,\ (\ :3/2),\ 0.447 - 0.894i)$

Particular Values

$L(2)$	$\approx$	$1.245673852$
$L(\frac12)$	$\approx$	$1.245673852$
$L(\frac{5}{2})$		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 + 3iT $
	5	$ 1 $
good	7	$ 1 - 5iT - 343T^{2} $
	11	$ 1 + 14T + 1.33e3T^{2} $
	13	$ 1 - iT - 2.19e3T^{2} $
	17	$ 1 + 46iT - 4.91e3T^{2} $
	19	$ 1 - 19T + 6.85e3T^{2} $
	23	$ 1 - 46iT - 1.21e4T^{2} $
	29	$ 1 + 14T + 2.43e4T^{2} $
	31	$ 1 + 133T + 2.97e4T^{2} $
	37	$ 1 + 258iT - 5.06e4T^{2} $
	41	$ 1 - 84T + 6.89e4T^{2} $
	43	$ 1 - 167iT - 7.95e4T^{2} $
	47	$ 1 - 410iT - 1.03e5T^{2} $
	53	$ 1 - 456iT - 1.48e5T^{2} $
	59	$ 1 + 194T + 2.05e5T^{2} $
	61	$ 1 + 17T + 2.26e5T^{2} $
	67	$ 1 - 653iT - 3.00e5T^{2} $
	71	$ 1 + 828T + 3.57e5T^{2} $
	73	$ 1 - 570iT - 3.89e5T^{2} $
	79	$ 1 + 552T + 4.93e5T^{2} $
	83	$ 1 + 142iT - 5.71e5T^{2} $
	89	$ 1 - 1.10e3T + 7.04e5T^{2} $
	97	$ 1 + 841iT - 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

−9.345351384744648606772442839705, −8.777134374669988223617351143887, −7.63425352151050579720158848178, −7.28703502884847903674895974005, −6.09493377238305412731079992222, −5.46025372065920046688909614585, −4.37812333010377834639422300518, −3.13964558403526928726489327418, −2.22803927767488163653189640446, −0.997453533672714072584271601255, 0.33226466449880231217591547039, 1.84938106257661067139180973486, 3.10458170886009886005081542092, 3.98829699584281758309677404845, 4.92064878766796327484429902369, 5.75688026059346262893877801932, 6.72043598227828869950645212363, 7.66770248606964398152314325281, 8.490078953783317211010443185000, 9.236608865731083763355389776747

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.245673852\)
\(L(\frac12)\)	\(\approx\)	\(1.245673852\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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