LMFDB - L-function 2-75-5.4-c17-0-0

Dirichlet series

L(s) = 1

− 65.1i·2^-s − 6.56e3i·3^-s + 1.26e5·4^-s − 4.27e5·6^-s + 2.28e7i·7^-s − 1.67e7i·8^-s − 4.30e7·9^-s − 5.40e8·11^-s − 8.32e8i·12^-s + 3.45e7i·13^-s + 1.48e9·14^-s + 1.55e10·16^-s − 9.43e9i·17^-s + 2.80e9i·18^-s + 2.25e9·19^-s + ⋯

L(s) = 1

− 0.179i·2^-s − 0.577i·3^-s + 0.967·4^-s − 0.103·6^-s + 1.49i·7^-s − 0.353i·8^-s − 0.333·9^-s − 0.759·11^-s − 0.558i·12^-s + 0.0117i·13^-s + 0.269·14^-s + 0.904·16^-s − 0.328i·17^-s + 0.0599i·18^-s + 0.0304·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(18-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$75$ = $3 \cdot 5^{2}$
Sign:	$-0.894 - 0.447i$
Analytic conductor:	$137.416$
Root analytic conductor:	$11.7224$
Motivic weight:	$17$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{75} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 75,\ (\ :17/2),\ -0.894 - 0.447i)$

Particular Values

$L(9)$	$\approx$	$0.3439424058$
$L(\frac12)$	$\approx$	$0.3439424058$
$L(\frac{19}{2})$		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 + 6.56e3iT $
	5	$ 1 $
good	2	$ 1 + 65.1iT - 1.31e5T^{2} $
	7	$ 1 - 2.28e7iT - 2.32e14T^{2} $
	11	$ 1 + 5.40e8T + 5.05e17T^{2} $
	13	$ 1 - 3.45e7iT - 8.65e18T^{2} $
	17	$ 1 + 9.43e9iT - 8.27e20T^{2} $
	19	$ 1 - 2.25e9T + 5.48e21T^{2} $
	23	$ 1 - 3.45e11iT - 1.41e23T^{2} $
	29	$ 1 + 5.11e11T + 7.25e24T^{2} $
	31	$ 1 - 1.14e11T + 2.25e25T^{2} $
	37	$ 1 + 1.56e13iT - 4.56e26T^{2} $
	41	$ 1 + 8.00e13T + 2.61e27T^{2} $
	43	$ 1 - 3.66e13iT - 5.87e27T^{2} $
	47	$ 1 + 1.17e14iT - 2.66e28T^{2} $
	53	$ 1 - 3.90e14iT - 2.05e29T^{2} $
	59	$ 1 + 1.82e15T + 1.27e30T^{2} $
	61	$ 1 - 1.41e15T + 2.24e30T^{2} $
	67	$ 1 + 1.47e15iT - 1.10e31T^{2} $
	71	$ 1 + 7.31e15T + 2.96e31T^{2} $
	73	$ 1 + 1.34e16iT - 4.74e31T^{2} $
	79	$ 1 - 8.37e15T + 1.81e32T^{2} $
	83	$ 1 - 2.55e16iT - 4.21e32T^{2} $
	89	$ 1 + 4.47e16T + 1.37e33T^{2} $
	97	$ 1 - 7.41e16iT - 5.95e33T^{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

−11.82755759815613370052352478018, −10.83435922915930763582244565551, −9.481458244076134068234971513998, −8.258402130235789295656458850527, −7.24872588741153052257146259112, −6.07788262425394028413110675221, −5.27082910065616474781515161038, −3.19849291994449255880117639074, −2.37160116115960980139615725150, −1.52440237776210627086701444196, 0.05873915200154127163503481477, 1.37286824065228699034837648216, 2.75316222318214366326603535075, 3.85549096330936112084398019100, 5.05163300447237961846522705937, 6.40515115920955940481325381899, 7.36598358669782451336423556656, 8.343624494842868224276213810519, 10.10794395674735663420236641585, 10.56925044905492330311852057869

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 $\F_{q}$
Belyi maps

\(L(9)\)	\(\approx\)	\(0.3439424058\)
\(L(\frac12)\)	\(\approx\)	\(0.3439424058\)
\(L(\frac{19}{2})\)		not available
\(L(1)\)		not available

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