LMFDB - L-function 2-75-15.14-c6-0-9

Dirichlet series

L(s) = 1

− 27i·3^-s − 64·4^-s + 286i·7^-s − 729·9^-s + 1.72e3i·12^-s + 506i·13^-s + 4.09e3·16^-s + 1.05e4·19^-s + 7.72e3·21^-s + 1.96e4i·27^-s − 1.83e4i·28^-s + 3.52e4·31^-s + 4.66e4·36^-s + 8.92e4i·37^-s + 1.36e4·39^-s + ⋯

L(s) = 1

− i·3^-s − 4^-s + 0.833i·7^-s − 0.999·9^-s + i·12^-s + 0.230i·13^-s + 16^-s + 1.54·19^-s + 0.833·21^-s + 0.999i·27^-s − 0.833i·28^-s + 1.18·31^-s + 0.999·36^-s + 1.76i·37^-s + 0.230·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(7-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$75$ = $3 \cdot 5^{2}$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$17.2540$
Root analytic conductor:	$4.15380$
Motivic weight:	$6$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{75} (74, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 75,\ (\ :3),\ 0.894 - 0.447i)$

Particular Values

$L(\frac{7}{2})$	$\approx$	$1.12414 + 0.265374i$
$L(\frac12)$	$\approx$	$1.12414 + 0.265374i$
$L(4)$		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 + 27iT $
	5	$ 1 $
good	2	$ 1 + 64T^{2} $
	7	$ 1 - 286iT - 1.17e5T^{2} $
	11	$ 1 - 1.77e6T^{2} $
	13	$ 1 - 506iT - 4.82e6T^{2} $
	17	$ 1 + 2.41e7T^{2} $
	19	$ 1 - 1.05e4T + 4.70e7T^{2} $
	23	$ 1 + 1.48e8T^{2} $
	29	$ 1 - 5.94e8T^{2} $
	31	$ 1 - 3.52e4T + 8.87e8T^{2} $
	37	$ 1 - 8.92e4iT - 2.56e9T^{2} $
	41	$ 1 - 4.75e9T^{2} $
	43	$ 1 - 1.11e5iT - 6.32e9T^{2} $
	47	$ 1 + 1.07e10T^{2} $
	53	$ 1 + 2.21e10T^{2} $
	59	$ 1 - 4.21e10T^{2} $
	61	$ 1 + 4.20e5T + 5.15e10T^{2} $
	67	$ 1 + 1.72e5iT - 9.04e10T^{2} $
	71	$ 1 - 1.28e11T^{2} $
	73	$ 1 - 6.38e5iT - 1.51e11T^{2} $
	79	$ 1 - 2.04e5T + 2.43e11T^{2} $
	83	$ 1 + 3.26e11T^{2} $
	89	$ 1 - 4.96e11T^{2} $
	97	$ 1 - 5.64e4iT - 8.32e11T^{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

−13.45768987889500577519856042939, −12.37682138915530620255834464222, −11.58208819437770289268308782701, −9.773930297832765476522475427445, −8.724339128276756680587988049490, −7.74370167361227050436387196653, −6.18036236124703964308565672939, −4.98282050826968890496107907231, −2.99090689549102054241367668730, −1.13585494306597733807108741887, 0.56162062531571553231000336270, 3.38256365635904452590860672978, 4.47441192740753461411610003764, 5.61024738007369046445097591718, 7.61802200503554585405045218592, 8.918601559971055287399416618298, 9.857817741933418514232921276411, 10.71927500835144163900893791338, 12.11307510215001099397436042564, 13.61587216363107662399393547845

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 $\F_{q}$

\(L(\frac{7}{2})\)	\(\approx\)	\(1.12414 + 0.265374i\)
\(L(\frac12)\)	\(\approx\)	\(1.12414 + 0.265374i\)
\(L(4)\)		not available
\(L(1)\)		not available

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