LMFDB - L-function 2-2151-717.716-c1-0-24

Dirichlet series

L(s) = 1

− 2.54i·2^-s − 4.45·4^-s + 2.74i·5^-s − 3.15i·7^-s + 6.23i·8^-s + 6.97·10^-s − 3.61i·11^-s + 6.68i·13^-s − 8.02·14^-s + 6.92·16^-s + 1.88i·17^-s − 1.30i·19^-s − 12.2i·20^-s − 9.19·22^-s − 0.0247·23^-s + ⋯

L(s) = 1

− 1.79i·2^-s − 2.22·4^-s + 1.22i·5^-s − 1.19i·7^-s + 2.20i·8^-s + 2.20·10^-s − 1.09i·11^-s + 1.85i·13^-s − 2.14·14^-s + 1.73·16^-s + 0.458i·17^-s − 0.299i·19^-s − 2.73i·20^-s − 1.95·22^-s − 0.00516·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0554 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0554 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2151$ = $3^{2} \cdot 239$
Sign:	$-0.0554 + 0.998i$
Analytic conductor:	$17.1758$
Root analytic conductor:	$4.14437$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2151} (2150, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2151,\ (\ :1/2),\ -0.0554 + 0.998i)$

Particular Values

$L(1)$	$\approx$	$1.387189349$
$L(\frac12)$	$\approx$	$1.387189349$
$L(\frac{3}{2})$		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 $
	239	$ 1 + (-13.0 + 8.21i)T $
good	2	$ 1 + 2.54iT - 2T^{2} $
	5	$ 1 - 2.74iT - 5T^{2} $
	7	$ 1 + 3.15iT - 7T^{2} $
	11	$ 1 + 3.61iT - 11T^{2} $
	13	$ 1 - 6.68iT - 13T^{2} $
	17	$ 1 - 1.88iT - 17T^{2} $
	19	$ 1 + 1.30iT - 19T^{2} $
	23	$ 1 + 0.0247T + 23T^{2} $
	29	$ 1 - 7.25iT - 29T^{2} $
	31	$ 1 - 0.823T + 31T^{2} $
	37	$ 1 + 7.67iT - 37T^{2} $
	41	$ 1 - 0.294T + 41T^{2} $
	43	$ 1 - 0.0550iT - 43T^{2} $
	47	$ 1 - 8.78T + 47T^{2} $
	53	$ 1 - 8.76T + 53T^{2} $
	59	$ 1 - 13.4T + 59T^{2} $
	61	$ 1 - 7.35T + 61T^{2} $
	67	$ 1 - 1.45T + 67T^{2} $
	71	$ 1 - 15.4iT - 71T^{2} $
	73	$ 1 + 0.0953iT - 73T^{2} $
	79	$ 1 + 12.3iT - 79T^{2} $
	83	$ 1 - 9.34iT - 83T^{2} $
	89	$ 1 - 3.09T + 89T^{2} $
	97	$ 1 + 12.1iT - 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

−9.095766529343238141467400531212, −8.517698909162481525809046000896, −7.16894120514923916898702041382, −6.76033429459262432688831064099, −5.48790874295355992675007277930, −4.12314379112393391564305505558, −3.88839788135924015683216195961, −2.93773550679765134260480093340, −2.03267622171494449941498878693, −0.865804933168855444867622519551, 0.70623470560726318322154748402, 2.48757536895815661660354550480, 4.03466470182501989130119721432, 4.98914695472453042919455921351, 5.36576001741309926175004602895, 5.97642486610013309132011981213, 6.97124756911873069832258836641, 7.933322253003712336410115027763, 8.225869976771817638914242270794, 8.986735541814359914331087582889

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.387189349\)
\(L(\frac12)\)	\(\approx\)	\(1.387189349\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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