LMFDB - L-function 2-4032-28.27-c1-0-77

Dirichlet series

L(s) = 1

− 3.33i·5^-s + (−1.56 − 2.13i)7^-s − 0.936i·11^-s + 1.87i·13^-s − 5.20i·17^-s − 7.12·19^-s − 0.936i·23^-s − 6.12·25^-s − 2·29^-s + (−7.12 + 5.20i)35^-s − 1.12·37^-s − 1.46i·41^-s + 9.06i·43^-s − 6.24·47^-s + (−2.12 + 6.67i)49^-s + ⋯

L(s) = 1

− 1.49i·5^-s + (−0.590 − 0.807i)7^-s − 0.282i·11^-s + 0.519i·13^-s − 1.26i·17^-s − 1.63·19^-s − 0.195i·23^-s − 1.22·25^-s − 0.371·29^-s + (−1.20 + 0.880i)35^-s − 0.184·37^-s − 0.228i·41^-s + 1.38i·43^-s − 0.911·47^-s + (−0.303 + 0.952i)49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$-0.590 - 0.807i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (3583, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ -0.590 - 0.807i)$

Particular Values

$L(1)$	$\approx$	$0.4342550393$
$L(\frac12)$	$\approx$	$0.4342550393$
$L(\frac{3}{2})$		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 $
	7	$ 1 + (1.56 + 2.13i)T $
good	5	$ 1 + 3.33iT - 5T^{2} $
	11	$ 1 + 0.936iT - 11T^{2} $
	13	$ 1 - 1.87iT - 13T^{2} $
	17	$ 1 + 5.20iT - 17T^{2} $
	19	$ 1 + 7.12T + 19T^{2} $
	23	$ 1 + 0.936iT - 23T^{2} $
	29	$ 1 + 2T + 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 1.12T + 37T^{2} $
	41	$ 1 + 1.46iT - 41T^{2} $
	43	$ 1 - 9.06iT - 43T^{2} $
	47	$ 1 + 6.24T + 47T^{2} $
	53	$ 1 - 12.2T + 53T^{2} $
	59	$ 1 - 4T + 59T^{2} $
	61	$ 1 - 4.79iT - 61T^{2} $
	67	$ 1 + 10.9iT - 67T^{2} $
	71	$ 1 + 3.86iT - 71T^{2} $
	73	$ 1 + 6.67iT - 73T^{2} $
	79	$ 1 - 2.39iT - 79T^{2} $
	83	$ 1 + 10.2T + 83T^{2} $
	89	$ 1 - 1.46iT - 89T^{2} $
	97	$ 1 - 10.4iT - 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

−8.102501829441555734965945823531, −7.21901780798345779129358488279, −6.52540462206281428422746651398, −5.71099643029847882081901224974, −4.72636724881449128168885727011, −4.36516226553020143976959940763, −3.42134106462095193158855587404, −2.21089357234790586987892699902, −1.05924310513774597501953971531, −0.13246836006065582480695438551, 1.94840963133941337043891174465, 2.58928783465127522574550550143, 3.47437646706531176724851991023, 4.13731957043879607625317647919, 5.47846367953164524373357654646, 6.05434328625753524647457986354, 6.71678611807216160606720266412, 7.24470900637021290786677025213, 8.332238742744967542820672921712, 8.711958046485518142988309105608

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.4342550393\)
\(L(\frac12)\)	\(\approx\)	\(0.4342550393\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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