LMFDB - L-function 2-5472-76.75-c1-0-52

Dirichlet series

L(s) = 1

− 2.62·5^-s + 3.45i·7^-s + 3.79i·11^-s − 0.312i·13^-s − 4.13·17^-s + (−3.41 + 2.70i)19^-s + 1.93i·23^-s + 1.87·25^-s − 5.99i·29^-s − 9.38·31^-s − 9.06i·35^-s + 6.64i·37^-s − 12.1i·41^-s + 10.2i·43^-s + 2.71i·47^-s + ⋯

L(s) = 1

− 1.17·5^-s + 1.30i·7^-s + 1.14i·11^-s − 0.0866i·13^-s − 1.00·17^-s + (−0.783 + 0.621i)19^-s + 0.402i·23^-s + 0.375·25^-s − 1.11i·29^-s − 1.68·31^-s − 1.53i·35^-s + 1.09i·37^-s − 1.89i·41^-s + 1.56i·43^-s + 0.396i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5472$ = $2^{5} \cdot 3^{2} \cdot 19$
Sign:	$-0.114 + 0.993i$
Analytic conductor:	$43.6941$
Root analytic conductor:	$6.61015$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5472} (2431, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5472,\ (\ :1/2),\ -0.114 + 0.993i)$

Particular Values

$L(1)$	$\approx$	$0.002916411417$
$L(\frac12)$	$\approx$	$0.002916411417$
$L(\frac{3}{2})$		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 $
	19	$ 1 + (3.41 - 2.70i)T $
good	5	$ 1 + 2.62T + 5T^{2} $
	7	$ 1 - 3.45iT - 7T^{2} $
	11	$ 1 - 3.79iT - 11T^{2} $
	13	$ 1 + 0.312iT - 13T^{2} $
	17	$ 1 + 4.13T + 17T^{2} $
	23	$ 1 - 1.93iT - 23T^{2} $
	29	$ 1 + 5.99iT - 29T^{2} $
	31	$ 1 + 9.38T + 31T^{2} $
	37	$ 1 - 6.64iT - 37T^{2} $
	41	$ 1 + 12.1iT - 41T^{2} $
	43	$ 1 - 10.2iT - 43T^{2} $
	47	$ 1 - 2.71iT - 47T^{2} $
	53	$ 1 - 7.29iT - 53T^{2} $
	59	$ 1 + 10.4T + 59T^{2} $
	61	$ 1 + 2.54T + 61T^{2} $
	67	$ 1 - 11.2T + 67T^{2} $
	71	$ 1 - 8.48T + 71T^{2} $
	73	$ 1 - 16.2T + 73T^{2} $
	79	$ 1 - 2.45T + 79T^{2} $
	83	$ 1 + 4.38iT - 83T^{2} $
	89	$ 1 + 5.30iT - 89T^{2} $
	97	$ 1 + 10.6iT - 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

−7.941440787987256089806621531923, −7.41358161019870099006964633684, −6.54664059484227265248597918874, −5.84945377801857968096388774063, −4.95427965190097804066947544748, −4.27537731310132173123293828133, −3.58007668325026996641641095908, −2.46292751806401177639798470940, −1.82036453614323173034086433078, −0.00104886240834221631665210936, 0.75233924418230965663344048548, 2.13881546355802204417811165724, 3.43844220952813284305324024066, 3.79587017716660917534617487103, 4.53327169378957214203067493873, 5.34109743219016972614394660699, 6.50532517134748406599213615175, 6.93893302119445967657777171854, 7.62886037685182964904497408721, 8.325355702550751504609724518712

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.002916411417\)
\(L(\frac12)\)	\(\approx\)	\(0.002916411417\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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