LMFDB - L-function 2-7200-12.11-c1-0-54

Dirichlet series

L(s) = 1

+ 3.50i·7^-s + 1.92·11^-s + 5.50·13^-s + 4.44i·17^-s − 7.00i·19^-s − 1.10·23^-s − 5.47i·29^-s − 8.28i·31^-s − 0.778·37^-s + 2.44i·41^-s − 9.55i·43^-s + 11.7·47^-s − 5.28·49^-s − 11.5i·53^-s + 9.78·59^-s + ⋯

L(s) = 1

+ 1.32i·7^-s + 0.581·11^-s + 1.52·13^-s + 1.07i·17^-s − 1.60i·19^-s − 0.229·23^-s − 1.01i·29^-s − 1.48i·31^-s − 0.127·37^-s + 0.381i·41^-s − 1.45i·43^-s + 1.70·47^-s − 0.754·49^-s − 1.58i·53^-s + 1.27·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.985 + 0.169i$
Analytic conductor:	$57.4922$
Root analytic conductor:	$7.58236$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7200} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7200,\ (\ :1/2),\ 0.985 + 0.169i)$

Particular Values

$L(1)$	$\approx$	$2.283338068$
$L(\frac12)$	$\approx$	$2.283338068$
$L(\frac{3}{2})$		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 - 3.50iT - 7T^{2} $
	11	$ 1 - 1.92T + 11T^{2} $
	13	$ 1 - 5.50T + 13T^{2} $
	17	$ 1 - 4.44iT - 17T^{2} $
	19	$ 1 + 7.00iT - 19T^{2} $
	23	$ 1 + 1.10T + 23T^{2} $
	29	$ 1 + 5.47iT - 29T^{2} $
	31	$ 1 + 8.28iT - 31T^{2} $
	37	$ 1 + 0.778T + 37T^{2} $
	41	$ 1 - 2.44iT - 41T^{2} $
	43	$ 1 + 9.55iT - 43T^{2} $
	47	$ 1 - 11.7T + 47T^{2} $
	53	$ 1 + 11.5iT - 53T^{2} $
	59	$ 1 - 9.78T + 59T^{2} $
	61	$ 1 + 3.45T + 61T^{2} $
	67	$ 1 + 5.45iT - 67T^{2} $
	71	$ 1 - 4.25T + 71T^{2} $
	73	$ 1 - 7.27T + 73T^{2} $
	79	$ 1 - 2.82iT - 79T^{2} $
	83	$ 1 + 4.25T + 83T^{2} $
	89	$ 1 - 0.386iT - 89T^{2} $
	97	$ 1 + 9.29T + 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

−8.169340404496542602646490197864, −7.11395239865003469397517978601, −6.32331676914324161065723791804, −5.91635998515549066148621061223, −5.24362698964024832541729431597, −4.16239978011497362965856121989, −3.66115765301413838762615554838, −2.53565360973572915128288983348, −1.93001550493765379385148257560, −0.67686663251443578730824041474, 1.00579027215786502311367617265, 1.45941152630257180054255717222, 2.92452767805097650727801757575, 3.78576304830741090607571494646, 4.08422041293013145337289221982, 5.12211969617220331940908280313, 5.92163867789072071535068245939, 6.63311678605543439705092492549, 7.21298594755334311748138490232, 7.87947245753074997037292284112

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.283338068\)
\(L(\frac12)\)	\(\approx\)	\(2.283338068\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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