LMFDB - L-function 2-675-15.14-c4-0-24

Dirichlet series

L(s) = 1

+ 3·2^-s − 7·4^-s − 19i·7^-s − 69·8^-s + 123i·11^-s − 302i·13^-s − 57i·14^-s − 95·16^-s − 414·17^-s + 304·19^-s + 369i·22^-s + 300·23^-s − 906i·26^-s + 133i·28^-s + 678i·29^-s + ⋯

L(s) = 1

+ 0.750·2^-s − 0.437·4^-s − 0.387i·7^-s − 1.07·8^-s + 1.01i·11^-s − 1.78i·13^-s − 0.290i·14^-s − 0.371·16^-s − 1.43·17^-s + 0.842·19^-s + 0.762i·22^-s + 0.567·23^-s − 1.34i·26^-s + 0.169i·28^-s + 0.806i·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(5-s) \end{aligned}\]

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

Invariants

Degree:	$2$
Conductor:	$675$ = $3^{3} \cdot 5^{2}$
Sign:	$0.447 - 0.894i$
Analytic conductor:	$69.7747$
Root analytic conductor:	$8.35312$
Motivic weight:	$4$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{675} (674, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 675,\ (\ :2),\ 0.447 - 0.894i)$

Particular Values

$L(\frac{5}{2})$	$\approx$	$1.668157359$
$L(\frac12)$	$\approx$	$1.668157359$
$L(3)$		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 - 3T + 16T^{2} $
	7	$ 1 + 19iT - 2.40e3T^{2} $
	11	$ 1 - 123iT - 1.46e4T^{2} $
	13	$ 1 + 302iT - 2.85e4T^{2} $
	17	$ 1 + 414T + 8.35e4T^{2} $
	19	$ 1 - 304T + 1.30e5T^{2} $
	23	$ 1 - 300T + 2.79e5T^{2} $
	29	$ 1 - 678iT - 7.07e5T^{2} $
	31	$ 1 - 239T + 9.23e5T^{2} $
	37	$ 1 - 740iT - 1.87e6T^{2} $
	41	$ 1 - 228iT - 2.82e6T^{2} $
	43	$ 1 - 982iT - 3.41e6T^{2} $
	47	$ 1 + 2.16e3T + 4.87e6T^{2} $
	53	$ 1 + 1.59e3T + 7.89e6T^{2} $
	59	$ 1 - 2.92e3iT - 1.21e7T^{2} $
	61	$ 1 + 316T + 1.38e7T^{2} $
	67	$ 1 - 4.62e3iT - 2.01e7T^{2} $
	71	$ 1 - 1.81e3iT - 2.54e7T^{2} $
	73	$ 1 - 3.03e3iT - 2.83e7T^{2} $
	79	$ 1 - 1.04e4T + 3.89e7T^{2} $
	83	$ 1 - 1.26e4T + 4.74e7T^{2} $
	89	$ 1 - 7.00e3iT - 6.27e7T^{2} $
	97	$ 1 + 6.51e3iT - 8.85e7T^{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

−10.07170849903350519232545446223, −9.262449874989938251825030416548, −8.318750056112559511531789804147, −7.36461434097534877319682821051, −6.38240100436307254072755977538, −5.24510192972590971729369197138, −4.69448249148793969122384676170, −3.58825513782979243998986641994, −2.64082826713242301691006409211, −0.914842229111898404844347337849, 0.38869769384841650663271258554, 2.08434433349122113068114062805, 3.30816371112308986368997291850, 4.26666875491485835832034767543, 5.06056392149586406666056878322, 6.10866149330715570789987416362, 6.78056204638243053838726511838, 8.173072083196391158003779548138, 9.139225282107012079173891217929, 9.345718570807899249536933071897

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

\(L(\frac{5}{2})\)	\(\approx\)	\(1.668157359\)
\(L(\frac12)\)	\(\approx\)	\(1.668157359\)
\(L(3)\)		not available
\(L(1)\)		not available

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