LMFDB - L-function 2-1656-69.68-c1-0-9

Dirichlet series

L(s) = 1

− 4.10·5^-s + 1.73i·7^-s + 3.47·11^-s − 4.27·13^-s − 5.54·17^-s + 4.56i·19^-s + (0.257 − 4.78i)23^-s + 11.8·25^-s − 7.63i·29^-s + 7.08·31^-s − 7.12i·35^-s + 1.02i·37^-s + 3.66i·41^-s + 0.721i·43^-s + 2.63i·47^-s + ⋯

L(s) = 1

− 1.83·5^-s + 0.657i·7^-s + 1.04·11^-s − 1.18·13^-s − 1.34·17^-s + 1.04i·19^-s + (0.0537 − 0.998i)23^-s + 2.36·25^-s − 1.41i·29^-s + 1.27·31^-s − 1.20i·35^-s + 0.167i·37^-s + 0.572i·41^-s + 0.109i·43^-s + 0.384i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1656$ = $2^{3} \cdot 3^{2} \cdot 23$
Sign:	$0.532 + 0.846i$
Analytic conductor:	$13.2232$
Root analytic conductor:	$3.63637$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1656} (1241, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1656,\ (\ :1/2),\ 0.532 + 0.846i)$

Particular Values

$L(1)$	$\approx$	$0.7560721612$
$L(\frac12)$	$\approx$	$0.7560721612$
$L(\frac{3}{2})$		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 $
	3	$ 1 $
	23	$ 1 + (-0.257 + 4.78i)T $
good	5	$ 1 + 4.10T + 5T^{2} $
	7	$ 1 - 1.73iT - 7T^{2} $
	11	$ 1 - 3.47T + 11T^{2} $
	13	$ 1 + 4.27T + 13T^{2} $
	17	$ 1 + 5.54T + 17T^{2} $
	19	$ 1 - 4.56iT - 19T^{2} $
	29	$ 1 + 7.63iT - 29T^{2} $
	31	$ 1 - 7.08T + 31T^{2} $
	37	$ 1 - 1.02iT - 37T^{2} $
	41	$ 1 - 3.66iT - 41T^{2} $
	43	$ 1 - 0.721iT - 43T^{2} $
	47	$ 1 - 2.63iT - 47T^{2} $
	53	$ 1 + 1.12T + 53T^{2} $
	59	$ 1 + 12.4iT - 59T^{2} $
	61	$ 1 + 11.1iT - 61T^{2} $
	67	$ 1 + 2.39iT - 67T^{2} $
	71	$ 1 + 0.495iT - 71T^{2} $
	73	$ 1 - 5.00T + 73T^{2} $
	79	$ 1 + 15.0iT - 79T^{2} $
	83	$ 1 - 15.2T + 83T^{2} $
	89	$ 1 - 2.94T + 89T^{2} $
	97	$ 1 - 1.86iT - 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

−9.120981510446664339726775095348, −8.277255553882578417441146881974, −7.86113806046332798293198353643, −6.83127655168617746421747043713, −6.24272300459892963147027941938, −4.71853568990289552857274204678, −4.34728163268761878898158910791, −3.33730130879705893868237287071, −2.22899729808473198433677411160, −0.39330186954548990049629848044, 0.900824170310170229435315754514, 2.67722484307343870281113703396, 3.78811484157324144678079589090, 4.33347543607996302738810583548, 5.09457115895706665665848076336, 6.79004071493308619025431937241, 7.03668796757263774548655737127, 7.77492230915893869796127416464, 8.748767878482404815542055220426, 9.250383513321883988227363009552

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7560721612\)
\(L(\frac12)\)	\(\approx\)	\(0.7560721612\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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