LMFDB - L-function 2-72e2-24.11-c1-0-68

Dirichlet series

L(s) = 1

+ 2.78·5^-s − 0.732i·7^-s − 2.03i·11^-s + 2.49i·13^-s + 1.55i·17^-s − 6.81·19^-s − 4.24·23^-s + 2.73·25^-s + 4.81·29^-s − 1.46i·31^-s − 2.03i·35^-s − 9.30i·37^-s − 7.34i·41^-s + 6.81·43^-s + 8.48·47^-s + ⋯

L(s) = 1

+ 1.24·5^-s − 0.276i·7^-s − 0.613i·11^-s + 0.691i·13^-s + 0.376i·17^-s − 1.56·19^-s − 0.884·23^-s + 0.546·25^-s + 0.894·29^-s − 0.262i·31^-s − 0.344i·35^-s − 1.52i·37^-s − 1.14i·41^-s + 1.03·43^-s + 1.23·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5184$ = $2^{6} \cdot 3^{4}$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$41.3944$
Root analytic conductor:	$6.43385$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5184} (2591, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5184,\ (\ :1/2),\ 0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$2.304164911$
$L(\frac12)$	$\approx$	$2.304164911$
$L(\frac{3}{2})$		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 $
good	5	$ 1 - 2.78T + 5T^{2} $
	7	$ 1 + 0.732iT - 7T^{2} $
	11	$ 1 + 2.03iT - 11T^{2} $
	13	$ 1 - 2.49iT - 13T^{2} $
	17	$ 1 - 1.55iT - 17T^{2} $
	19	$ 1 + 6.81T + 19T^{2} $
	23	$ 1 + 4.24T + 23T^{2} $
	29	$ 1 - 4.81T + 29T^{2} $
	31	$ 1 + 1.46iT - 31T^{2} $
	37	$ 1 + 9.30iT - 37T^{2} $
	41	$ 1 + 7.34iT - 41T^{2} $
	43	$ 1 - 6.81T + 43T^{2} $
	47	$ 1 - 8.48T + 47T^{2} $
	53	$ 1 - 7.59T + 53T^{2} $
	59	$ 1 + 9.63iT - 59T^{2} $
	61	$ 1 - 4.31iT - 61T^{2} $
	67	$ 1 - 11.7T + 67T^{2} $
	71	$ 1 - 7.34T + 71T^{2} $
	73	$ 1 - 6.66T + 73T^{2} $
	79	$ 1 + 6.19iT - 79T^{2} $
	83	$ 1 + 1.49iT - 83T^{2} $
	89	$ 1 - 8.90iT - 89T^{2} $
	97	$ 1 - 8.92T + 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

−8.286634647691763103991307093208, −7.29599195671136428068580560904, −6.54725918407574960510983769705, −5.95470098832251405259706360535, −5.45219966344183878739729994969, −4.28095861219127624656501384642, −3.79967698237964296940189647573, −2.34079825346828900064866610041, −2.04443593971658192179075023754, −0.65958065435556590809555704431, 1.04486121360685577032221400055, 2.23567803673306832480824757867, 2.58272366383735200289021347876, 3.88877757847353951731874431880, 4.74141510781047344306335263406, 5.45726760827306326138978820778, 6.19996153392924600906680877983, 6.63055676881134036042780599625, 7.62644240022588290062277212045, 8.381846841611226014725736722473

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.304164911\)
\(L(\frac12)\)	\(\approx\)	\(2.304164911\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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