LMFDB - L-function 2-20e2-5.4-c5-0-28

Dirichlet series

L(s) = 1

+ 18i·3^-s + 242i·7^-s − 81·9^-s − 656·11^-s − 206i·13^-s − 1.69e3i·17^-s − 1.36e3·19^-s − 4.35e3·21^-s − 2.19e3i·23^-s + 2.91e3i·27^-s + 2.21e3·29^-s + 1.70e3·31^-s − 1.18e4i·33^-s + 846i·37^-s + 3.70e3·39^-s + ⋯

L(s) = 1

+ 1.15i·3^-s + 1.86i·7^-s − 0.333·9^-s − 1.63·11^-s − 0.338i·13^-s − 1.41i·17^-s − 0.866·19^-s − 2.15·21^-s − 0.866i·23^-s + 0.769i·27^-s + 0.489·29^-s + 0.317·31^-s − 1.88i·33^-s + 0.101i·37^-s + 0.390·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$400$ = $2^{4} \cdot 5^{2}$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$64.1535$
Root analytic conductor:	$8.00958$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{400} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 400,\ (\ :5/2),\ 0.447 + 0.894i)$

Particular Values

$L(3)$	$\approx$	$0.2265877557$
$L(\frac12)$	$\approx$	$0.2265877557$
$L(\frac{7}{2})$		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 $
	5	$ 1 $
good	3	$ 1 - 18iT - 243T^{2} $
	7	$ 1 - 242iT - 1.68e4T^{2} $
	11	$ 1 + 656T + 1.61e5T^{2} $
	13	$ 1 + 206iT - 3.71e5T^{2} $
	17	$ 1 + 1.69e3iT - 1.41e6T^{2} $
	19	$ 1 + 1.36e3T + 2.47e6T^{2} $
	23	$ 1 + 2.19e3iT - 6.43e6T^{2} $
	29	$ 1 - 2.21e3T + 2.05e7T^{2} $
	31	$ 1 - 1.70e3T + 2.86e7T^{2} $
	37	$ 1 - 846iT - 6.93e7T^{2} $
	41	$ 1 + 1.81e3T + 1.15e8T^{2} $
	43	$ 1 + 1.05e4iT - 1.47e8T^{2} $
	47	$ 1 - 1.20e4iT - 2.29e8T^{2} $
	53	$ 1 - 3.25e4iT - 4.18e8T^{2} $
	59	$ 1 - 8.66e3T + 7.14e8T^{2} $
	61	$ 1 + 3.46e4T + 8.44e8T^{2} $
	67	$ 1 + 4.75e4iT - 1.35e9T^{2} $
	71	$ 1 + 948T + 1.80e9T^{2} $
	73	$ 1 + 6.31e4iT - 2.07e9T^{2} $
	79	$ 1 - 4.65e4T + 3.07e9T^{2} $
	83	$ 1 - 8.87e4iT - 3.93e9T^{2} $
	89	$ 1 - 1.04e5T + 5.58e9T^{2} $
	97	$ 1 - 3.62e4iT - 8.58e9T^{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

−10.31172682188460086486590354602, −9.359928473156736680533288580607, −8.712647417952766837336505650084, −7.73979829652982475043249870085, −6.20490876722207459597183547641, −5.19302384307400839565448288388, −4.71916898369486766200969715915, −3.02133857873629167232537570509, −2.37443700297175720114766246713, −0.05943118198874736705756973704, 1.05551199823078966696348272244, 2.09997337865011523416872817435, 3.63808181675809306608218983480, 4.71120428365409301494224423758, 6.16090953726794671604024093392, 7.01720508655105318316414923753, 7.74230748727845774518254614886, 8.334856608351929095004169391129, 10.09996619736733656994003085014, 10.48412043892979570426331620970

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(0.2265877557\)
\(L(\frac12)\)	\(\approx\)	\(0.2265877557\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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