LMFDB - L-function 2-7200-40.29-c1-0-87

Dirichlet series

L(s) = 1

− 4.72i·7^-s − 3.93i·11^-s − 3.46·13^-s − 3.51i·17^-s − 5.44i·19^-s + 7.11i·23^-s − 3.66i·29^-s − 5.23·31^-s + 0.414·37^-s − 3.00·41^-s − 5.34·43^-s − 0.925i·47^-s − 15.3·49^-s + 0.233·53^-s − 14.3i·59^-s + ⋯

L(s) = 1

− 1.78i·7^-s − 1.18i·11^-s − 0.961·13^-s − 0.852i·17^-s − 1.24i·19^-s + 1.48i·23^-s − 0.681i·29^-s − 0.940·31^-s + 0.0681·37^-s − 0.469·41^-s − 0.815·43^-s − 0.135i·47^-s − 2.18·49^-s + 0.0320·53^-s − 1.87i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.873 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.873 - 0.487i$
Analytic conductor:	$57.4922$
Root analytic conductor:	$7.58236$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7200} (2449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7200,\ (\ :1/2),\ -0.873 - 0.487i)$

Particular Values

$L(1)$	$\approx$	$0.8594922372$
$L(\frac12)$	$\approx$	$0.8594922372$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
good	7	$ 1 + 4.72iT - 7T^{2} $
	11	$ 1 + 3.93iT - 11T^{2} $
	13	$ 1 + 3.46T + 13T^{2} $
	17	$ 1 + 3.51iT - 17T^{2} $
	19	$ 1 + 5.44iT - 19T^{2} $
	23	$ 1 - 7.11iT - 23T^{2} $
	29	$ 1 + 3.66iT - 29T^{2} $
	31	$ 1 + 5.23T + 31T^{2} $
	37	$ 1 - 0.414T + 37T^{2} $
	41	$ 1 + 3.00T + 41T^{2} $
	43	$ 1 + 5.34T + 43T^{2} $
	47	$ 1 + 0.925iT - 47T^{2} $
	53	$ 1 - 0.233T + 53T^{2} $
	59	$ 1 + 14.3iT - 59T^{2} $
	61	$ 1 + 0.118iT - 61T^{2} $
	67	$ 1 - 13.4T + 67T^{2} $
	71	$ 1 - 2.19T + 71T^{2} $
	73	$ 1 - 0.563iT - 73T^{2} $
	79	$ 1 + 10.2T + 79T^{2} $
	83	$ 1 - 11.3T + 83T^{2} $
	89	$ 1 - 8.88T + 89T^{2} $
	97	$ 1 - 7.27iT - 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

−7.43560919760727797807941825165, −6.99591451826480860998274489313, −6.26826312924002707123932975640, −5.16928822287008644789733944353, −4.80452429816944822196805438676, −3.69857969691295126491260616422, −3.33876804011047959357824002590, −2.19548091083463118433549845672, −0.942917653022729256911512663616, −0.23025832874562496155719786865, 1.73511485638872602793777028120, 2.21258602951661320446171398283, 3.05273459215403327639841145379, 4.10222937043102162517085514565, 4.91740306190989553880213653385, 5.48343309007010654188074981584, 6.19194857157089017044610252647, 6.88539259191103020505459438847, 7.69079254235678835266494274914, 8.431878840311567190742706550243

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8594922372\)
\(L(\frac12)\)	\(\approx\)	\(0.8594922372\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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