LMFDB - L-function 2-7200-40.29-c1-0-25

Dirichlet series

L(s) = 1

− 0.0802i·7^-s − 2.41i·11^-s + 5.26·13^-s + 0.255i·17^-s + 6.95i·19^-s + 1.64i·23^-s + 4.51i·29^-s − 8.29·31^-s − 2.67·37^-s + 8.11·41^-s + 4.08·43^-s + 5.70i·47^-s + 6.99·49^-s − 11.5·53^-s − 12.6i·59^-s + ⋯

L(s) = 1

− 0.0303i·7^-s − 0.728i·11^-s + 1.46·13^-s + 0.0620i·17^-s + 1.59i·19^-s + 0.343i·23^-s + 0.838i·29^-s − 1.48·31^-s − 0.439·37^-s + 1.26·41^-s + 0.623·43^-s + 0.831i·47^-s + 0.999·49^-s − 1.58·53^-s − 1.65i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.321 - 0.947i$
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:	$(2,\ 7200,\ (\ :1/2),\ 0.321 - 0.947i)$

Particular Values

$L(1)$	$\approx$	$1.757478517$
$L(\frac12)$	$\approx$	$1.757478517$
$L(\frac{3}{2})$		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 + 0.0802iT - 7T^{2} $
	11	$ 1 + 2.41iT - 11T^{2} $
	13	$ 1 - 5.26T + 13T^{2} $
	17	$ 1 - 0.255iT - 17T^{2} $
	19	$ 1 - 6.95iT - 19T^{2} $
	23	$ 1 - 1.64iT - 23T^{2} $
	29	$ 1 - 4.51iT - 29T^{2} $
	31	$ 1 + 8.29T + 31T^{2} $
	37	$ 1 + 2.67T + 37T^{2} $
	41	$ 1 - 8.11T + 41T^{2} $
	43	$ 1 - 4.08T + 43T^{2} $
	47	$ 1 - 5.70iT - 47T^{2} $
	53	$ 1 + 11.5T + 53T^{2} $
	59	$ 1 + 12.6iT - 59T^{2} $
	61	$ 1 - 11.9iT - 61T^{2} $
	67	$ 1 + 7.27T + 67T^{2} $
	71	$ 1 + 11.3T + 71T^{2} $
	73	$ 1 - 12.0iT - 73T^{2} $
	79	$ 1 - 5.50T + 79T^{2} $
	83	$ 1 + 9.20T + 83T^{2} $
	89	$ 1 - 11.9T + 89T^{2} $
	97	$ 1 - 8.50iT - 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

−8.000308552564316506423034344995, −7.51561132995590053073484018740, −6.55800302278402762127011117542, −5.81364813890264942500410772908, −5.57747432580025131909160144914, −4.32491278284476199756820609636, −3.66341271164760470028524454024, −3.10379060866524906720683417268, −1.80844503233899297886087963571, −1.07171897113512233319299201976, 0.46618433199657005086950150046, 1.62668509660903625740424530744, 2.51180426547800018661815031044, 3.42708046924629458267138494371, 4.24148745156041527664270440677, 4.85378709018428729830376888118, 5.80958145346889752437888302638, 6.31914890801897923391037250236, 7.21213050863825829926652682956, 7.61320168039796886992603414747

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.757478517\)
\(L(\frac12)\)	\(\approx\)	\(1.757478517\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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