LMFDB - L-function 2-39e2-13.12-c1-0-56

Dirichlet series

L(s) = 1

− 2.35i·2^-s − 3.55·4^-s − 3.69i·5^-s + 0.801i·7^-s + 3.66i·8^-s − 8.70·10^-s − 2.85i·11^-s + 1.89·14^-s + 1.52·16^-s + 2.93·17^-s − 2.44i·19^-s + 13.1i·20^-s − 6.71·22^-s − 7.78·23^-s − 8.63·25^-s + ⋯

L(s) = 1

− 1.66i·2^-s − 1.77·4^-s − 1.65i·5^-s + 0.303i·7^-s + 1.29i·8^-s − 2.75·10^-s − 0.859i·11^-s + 0.505·14^-s + 0.381·16^-s + 0.712·17^-s − 0.560i·19^-s + 2.93i·20^-s − 1.43·22^-s − 1.62·23^-s − 1.72·25^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1521$ = $3^{2} \cdot 13^{2}$
Sign:	$0.246 - 0.969i$
Analytic conductor:	$12.1452$
Root analytic conductor:	$3.48500$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1521} (1351, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1521,\ (\ :1/2),\ 0.246 - 0.969i)$

Particular Values

$L(1)$	$\approx$	$0.9359201103$
$L(\frac12)$	$\approx$	$0.9359201103$
$L(\frac{3}{2})$		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 $
	13	$ 1 $
good	2	$ 1 + 2.35iT - 2T^{2} $
	5	$ 1 + 3.69iT - 5T^{2} $
	7	$ 1 - 0.801iT - 7T^{2} $
	11	$ 1 + 2.85iT - 11T^{2} $
	17	$ 1 - 2.93T + 17T^{2} $
	19	$ 1 + 2.44iT - 19T^{2} $
	23	$ 1 + 7.78T + 23T^{2} $
	29	$ 1 + 3.85T + 29T^{2} $
	31	$ 1 - 2.34iT - 31T^{2} $
	37	$ 1 + 7.44iT - 37T^{2} $
	41	$ 1 - 0.850iT - 41T^{2} $
	43	$ 1 - 1.61T + 43T^{2} $
	47	$ 1 - 2.44iT - 47T^{2} $
	53	$ 1 - 9.96T + 53T^{2} $
	59	$ 1 + 5.38iT - 59T^{2} $
	61	$ 1 + 13.2T + 61T^{2} $
	67	$ 1 - 14.3iT - 67T^{2} $
	71	$ 1 + 8.12iT - 71T^{2} $
	73	$ 1 - 11.8iT - 73T^{2} $
	79	$ 1 - 5.40T + 79T^{2} $
	83	$ 1 + 7.04iT - 83T^{2} $
	89	$ 1 - 1.13iT - 89T^{2} $
	97	$ 1 + 5.94iT - 97T^{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

−8.966864587670790561553631526170, −8.558493824372101891413849296708, −7.60292297205667884567578696711, −5.92406443172446175622632301933, −5.27389846330160119624110439135, −4.30613812988014028224377894392, −3.63363874953090742454773412152, −2.40493117856235023151989027897, −1.37316367862143324126032901249, −0.38219061254588575890010683466, 2.17104215548601367344215788341, 3.53081691537283655566765211294, 4.36080469016562574588959453430, 5.57819231590894927796438893392, 6.22550411743112477934166812174, 6.88879179985235424258210185718, 7.64303699838596872026951589921, 7.909316964484153528692709518387, 9.189833052945895034285577700176, 10.05651932795061784639571952774

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9359201103\)
\(L(\frac12)\)	\(\approx\)	\(0.9359201103\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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