LMFDB - L-function 2-15e2-5.4-c9-0-13

Dirichlet series

L(s) = 1

+ 43.8i·2^-s − 1.41e3·4^-s + 7.86e3i·7^-s − 3.95e4i·8^-s + 4.93e4·11^-s + 2.42e4i·13^-s − 3.44e5·14^-s + 1.01e6·16^-s + 2.68e5i·17^-s + 1.68e5·19^-s + 2.16e6i·22^-s + 2.12e6i·23^-s − 1.06e6·26^-s − 1.11e7i·28^-s + 3.89e5·29^-s + ⋯

L(s) = 1

+ 1.93i·2^-s − 2.76·4^-s + 1.23i·7^-s − 3.41i·8^-s + 1.01·11^-s + 0.235i·13^-s − 2.40·14^-s + 3.86·16^-s + 0.778i·17^-s + 0.296·19^-s + 1.97i·22^-s + 1.58i·23^-s − 0.456·26^-s − 3.41i·28^-s + 0.102·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(10-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$225$ = $3^{2} \cdot 5^{2}$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$115.883$
Root analytic conductor:	$10.7648$
Motivic weight:	$9$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{225} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 225,\ (\ :9/2),\ 0.447 + 0.894i)$

Particular Values

$L(5)$	$\approx$	$1.282454665$
$L(\frac12)$	$\approx$	$1.282454665$
$L(\frac{11}{2})$		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 $
	5	$ 1 $
good	2	$ 1 - 43.8iT - 512T^{2} $
	7	$ 1 - 7.86e3iT - 4.03e7T^{2} $
	11	$ 1 - 4.93e4T + 2.35e9T^{2} $
	13	$ 1 - 2.42e4iT - 1.06e10T^{2} $
	17	$ 1 - 2.68e5iT - 1.18e11T^{2} $
	19	$ 1 - 1.68e5T + 3.22e11T^{2} $
	23	$ 1 - 2.12e6iT - 1.80e12T^{2} $
	29	$ 1 - 3.89e5T + 1.45e13T^{2} $
	31	$ 1 - 9.05e4T + 2.64e13T^{2} $
	37	$ 1 - 3.31e6iT - 1.29e14T^{2} $
	41	$ 1 + 2.32e7T + 3.27e14T^{2} $
	43	$ 1 - 1.91e7iT - 5.02e14T^{2} $
	47	$ 1 - 6.28e7iT - 1.11e15T^{2} $
	53	$ 1 - 1.80e5iT - 3.29e15T^{2} $
	59	$ 1 - 3.84e7T + 8.66e15T^{2} $
	61	$ 1 + 5.53e5T + 1.16e16T^{2} $
	67	$ 1 - 2.39e8iT - 2.72e16T^{2} $
	71	$ 1 + 1.28e8T + 4.58e16T^{2} $
	73	$ 1 + 2.39e8iT - 5.88e16T^{2} $
	79	$ 1 - 5.28e8T + 1.19e17T^{2} $
	83	$ 1 + 2.12e8iT - 1.86e17T^{2} $
	89	$ 1 + 2.07e8T + 3.50e17T^{2} $
	97	$ 1 + 1.70e9iT - 7.60e17T^{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

−11.62345794977554462793838586540, −9.799595082567991453588098082200, −9.097637279055044360378702711868, −8.381210380720700311687829300210, −7.36697470864644085938755213570, −6.30519103577572713151334370894, −5.71321097447361480522226150588, −4.65642657817789885464537507733, −3.47364006490433815397166686034, −1.37214266540901564084775857721, 0.34523772384080472534054126602, 0.971971180180974662979920835451, 2.14205520848433895364177083933, 3.39042209510176586209561939356, 4.13007854635118792960722248376, 5.08826264288552096409262559836, 6.87219043576745223185795965241, 8.291034234794087585322307806674, 9.237470579641770868613313882891, 10.15994624911894525568852105483

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 $\F_{q}$

\(L(5)\)	\(\approx\)	\(1.282454665\)
\(L(\frac12)\)	\(\approx\)	\(1.282454665\)
\(L(\frac{11}{2})\)		not available
\(L(1)\)		not available

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