LMFDB - L-function 2-30e2-12.11-c1-0-15

Dirichlet series

L(s) = 1

− 1.41i·2^-s − 2.00·4^-s + 2.82i·8^-s + 6·13^-s + 4.00·16^-s + 7.07i·17^-s − 8.48i·26^-s − 4.24i·29^-s − 5.65i·32^-s + 10.0·34^-s + 12·37^-s − 12.7i·41^-s + 7·49^-s − 12.0·52^-s + 7.07i·53^-s + ⋯

L(s) = 1

− 0.999i·2^-s − 1.00·4^-s + 1.00i·8^-s + 1.66·13^-s + 1.00·16^-s + 1.71i·17^-s − 1.66i·26^-s − 0.787i·29^-s − 1.00i·32^-s + 1.71·34^-s + 1.97·37^-s − 1.98i·41^-s + 49^-s − 1.66·52^-s + 0.971i·53^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$900$ = $2^{2} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.577 + 0.816i$
Analytic conductor:	$7.18653$
Root analytic conductor:	$2.68077$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{900} (251, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 900,\ (\ :1/2),\ 0.577 + 0.816i)$

Particular Values

$L(1)$	$\approx$	$1.31801 - 0.682255i$
$L(\frac12)$	$\approx$	$1.31801 - 0.682255i$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 + 1.41iT $
	3	$ 1 $
	5	$ 1 $
good	7	$ 1 - 7T^{2} $
	11	$ 1 + 11T^{2} $
	13	$ 1 - 6T + 13T^{2} $
	17	$ 1 - 7.07iT - 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 + 4.24iT - 29T^{2} $
	31	$ 1 - 31T^{2} $
	37	$ 1 - 12T + 37T^{2} $
	41	$ 1 + 12.7iT - 41T^{2} $
	43	$ 1 - 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 7.07iT - 53T^{2} $
	59	$ 1 + 59T^{2} $
	61	$ 1 - 10T + 61T^{2} $
	67	$ 1 - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 6T + 73T^{2} $
	79	$ 1 - 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 - 4.24iT - 89T^{2} $
	97	$ 1 - 18T + 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

−10.25112595871549165988828615303, −9.155660799531239962095627864566, −8.527370564520796374786319775287, −7.77503314264071377378132268493, −6.25816860652502434683774886375, −5.62241557199694634271569931150, −4.17818978083096524497822972411, −3.69370646875757852353872903048, −2.31461256321430068668151782689, −1.10223793934341277634574173736, 0.982513810472155264320789039651, 3.03272030203649450494233059549, 4.14603486135789837797135601072, 5.09844886409146014489383588442, 6.02599003785782774507689382753, 6.77090012924321464105670294857, 7.67451785925973910509406500315, 8.479424804906919522049597181380, 9.227155808484390333312558959310, 9.954671231493981989028004362702

Overview	Random
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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

\(L(1)\)	\(\approx\)	\(1.31801 - 0.682255i\)
\(L(\frac12)\)	\(\approx\)	\(1.31801 - 0.682255i\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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