LMFDB - L-function 2-825-5.4-c3-0-40

Dirichlet series

L(s) = 1

− 5i·2^-s + 3i·3^-s − 17·4^-s + 15·6^-s − 3i·7^-s + 45i·8^-s − 9·9^-s − 11·11^-s − 51i·12^-s + 32i·13^-s − 15·14^-s + 89·16^-s − 33i·17^-s + 45i·18^-s − 47·19^-s + ⋯

L(s) = 1

− 1.76i·2^-s + 0.577i·3^-s − 2.12·4^-s + 1.02·6^-s − 0.161i·7^-s + 1.98i·8^-s − 0.333·9^-s − 0.301·11^-s − 1.22i·12^-s + 0.682i·13^-s − 0.286·14^-s + 1.39·16^-s − 0.470i·17^-s + 0.589i·18^-s − 0.567·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$825$ = $3 \cdot 5^{2} \cdot 11$
Sign:	$-0.447 + 0.894i$
Analytic conductor:	$48.6765$
Root analytic conductor:	$6.97686$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{825} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 825,\ (\ :3/2),\ -0.447 + 0.894i)$

Particular Values

$L(2)$	$\approx$	$1.410612480$
$L(\frac12)$	$\approx$	$1.410612480$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 - 3iT $
	5	$ 1 $
	11	$ 1 + 11T $
good	2	$ 1 + 5iT - 8T^{2} $
	7	$ 1 + 3iT - 343T^{2} $
	13	$ 1 - 32iT - 2.19e3T^{2} $
	17	$ 1 + 33iT - 4.91e3T^{2} $
	19	$ 1 + 47T + 6.85e3T^{2} $
	23	$ 1 - 113iT - 1.21e4T^{2} $
	29	$ 1 - 54T + 2.43e4T^{2} $
	31	$ 1 - 178T + 2.97e4T^{2} $
	37	$ 1 + 19iT - 5.06e4T^{2} $
	41	$ 1 - 139T + 6.89e4T^{2} $
	43	$ 1 + 308iT - 7.95e4T^{2} $
	47	$ 1 + 195iT - 1.03e5T^{2} $
	53	$ 1 - 152iT - 1.48e5T^{2} $
	59	$ 1 - 625T + 2.05e5T^{2} $
	61	$ 1 - 320T + 2.26e5T^{2} $
	67	$ 1 + 200iT - 3.00e5T^{2} $
	71	$ 1 + 947T + 3.57e5T^{2} $
	73	$ 1 + 448iT - 3.89e5T^{2} $
	79	$ 1 - 721T + 4.93e5T^{2} $
	83	$ 1 - 142iT - 5.71e5T^{2} $
	89	$ 1 + 404T + 7.04e5T^{2} $
	97	$ 1 + 79iT - 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

−9.841139287661426710788325195885, −9.090754077471905503795104381851, −8.360905321155435004219060395628, −7.04892339545878997809637865298, −5.61325758349549965541769990694, −4.60620891709721076657122269766, −3.91584705611531397102102537421, −2.91973842983730435902445869112, −1.94195649750244010518997734865, −0.58959185299038811259206839098, 0.73231626125972804485249840533, 2.60213957766508378798122485212, 4.16645588046454877189034600445, 5.12592780607657543352540847346, 6.02899905503742574519540327488, 6.58501267153422060534714171772, 7.51150433612711306487467253938, 8.252021314928925186019479405980, 8.699171779132760610936321761552, 9.842417329751743445817387824533

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.410612480\)
\(L(\frac12)\)	\(\approx\)	\(1.410612480\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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