LMFDB - L-function 2-675-5.4-c3-0-38

Dirichlet series

L(s) = 1

+ 4.45i·2^-s − 11.8·4^-s − 5.08i·7^-s − 17.3i·8^-s − 58.3·11^-s + 21.2i·13^-s + 22.6·14^-s − 17.8·16^-s − 68.8i·17^-s + 40.8·19^-s − 259. i·22^-s + 144. i·23^-s − 94.5·26^-s + 60.3i·28^-s − 220.·29^-s + ⋯

L(s) = 1

+ 1.57i·2^-s − 1.48·4^-s − 0.274i·7^-s − 0.765i·8^-s − 1.59·11^-s + 0.452i·13^-s + 0.432·14^-s − 0.278·16^-s − 0.982i·17^-s + 0.492·19^-s − 2.51i·22^-s + 1.30i·23^-s − 0.713·26^-s + 0.407i·28^-s − 1.40·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$675$ = $3^{3} \cdot 5^{2}$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$39.8262$
Root analytic conductor:	$6.31080$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{675} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 675,\ (\ :3/2),\ 0.894 - 0.447i)$

Particular Values

$L(2)$	$\approx$	$0.9943677373$
$L(\frac12)$	$\approx$	$0.9943677373$
$L(\frac{5}{2})$		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 $
	5	$ 1 $
good	2	$ 1 - 4.45iT - 8T^{2} $
	7	$ 1 + 5.08iT - 343T^{2} $
	11	$ 1 + 58.3T + 1.33e3T^{2} $
	13	$ 1 - 21.2iT - 2.19e3T^{2} $
	17	$ 1 + 68.8iT - 4.91e3T^{2} $
	19	$ 1 - 40.8T + 6.85e3T^{2} $
	23	$ 1 - 144. iT - 1.21e4T^{2} $
	29	$ 1 + 220.T + 2.43e4T^{2} $
	31	$ 1 - 291.T + 2.97e4T^{2} $
	37	$ 1 + 260. iT - 5.06e4T^{2} $
	41	$ 1 - 169.T + 6.89e4T^{2} $
	43	$ 1 + 438. iT - 7.95e4T^{2} $
	47	$ 1 + 255. iT - 1.03e5T^{2} $
	53	$ 1 + 214. iT - 1.48e5T^{2} $
	59	$ 1 - 331.T + 2.05e5T^{2} $
	61	$ 1 - 54.9T + 2.26e5T^{2} $
	67	$ 1 + 758. iT - 3.00e5T^{2} $
	71	$ 1 - 904.T + 3.57e5T^{2} $
	73	$ 1 - 866. iT - 3.89e5T^{2} $
	79	$ 1 + 206.T + 4.93e5T^{2} $
	83	$ 1 - 463. iT - 5.71e5T^{2} $
	89	$ 1 - 601.T + 7.04e5T^{2} $
	97	$ 1 + 229. iT - 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

−9.851334404736317357671774432569, −9.107669891487995054135097420324, −8.057951900375952081075878177929, −7.46817422580999530575202340170, −6.83735562271273727985228382471, −5.50624884525219354561002452778, −5.22948159504429983294118847357, −3.92675635240559193458960460843, −2.40809580187535808164944235331, −0.33371892252096111230374058078, 0.982483936409185624519269565670, 2.37800815811876254332999011334, 2.99436100914583143567323470228, 4.23986839540833921490792650259, 5.17662748439117658362962948428, 6.29613099818038788599820377926, 7.76948005825038946831702015514, 8.460753148221496131511489503826, 9.544791872155365912946285047456, 10.32095619040348418012261483951

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.9943677373\)
\(L(\frac12)\)	\(\approx\)	\(0.9943677373\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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