LMFDB - L-function 2-7200-8.5-c1-0-81

Dirichlet series

L(s) = 1

− 3.62·7^-s − 6.20i·11^-s − 0.578i·13^-s + 1.42·17^-s − 5.62i·19^-s + 5.62·23^-s + 2i·29^-s + 2.57·31^-s − 7.83i·37^-s − 5.25·41^-s + 7.25i·43^-s − 6.78·47^-s + 6.15·49^-s + 2i·53^-s + 2.20i·59^-s + ⋯

L(s) = 1

− 1.37·7^-s − 1.87i·11^-s − 0.160i·13^-s + 0.344·17^-s − 1.29i·19^-s + 1.17·23^-s + 0.371i·29^-s + 0.463·31^-s − 1.28i·37^-s − 0.820·41^-s + 1.10i·43^-s − 0.989·47^-s + 0.879·49^-s + 0.274i·53^-s + 0.287i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.994 + 0.102i$
Analytic conductor:	$57.4922$
Root analytic conductor:	$7.58236$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7200} (3601, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7200,\ (\ :1/2),\ -0.994 + 0.102i)$

Particular Values

$L(1)$	$\approx$	$0.7463609261$
$L(\frac12)$	$\approx$	$0.7463609261$
$L(\frac{3}{2})$		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 + 3.62T + 7T^{2} $
	11	$ 1 + 6.20iT - 11T^{2} $
	13	$ 1 + 0.578iT - 13T^{2} $
	17	$ 1 - 1.42T + 17T^{2} $
	19	$ 1 + 5.62iT - 19T^{2} $
	23	$ 1 - 5.62T + 23T^{2} $
	29	$ 1 - 2iT - 29T^{2} $
	31	$ 1 - 2.57T + 31T^{2} $
	37	$ 1 + 7.83iT - 37T^{2} $
	41	$ 1 + 5.25T + 41T^{2} $
	43	$ 1 - 7.25iT - 43T^{2} $
	47	$ 1 + 6.78T + 47T^{2} $
	53	$ 1 - 2iT - 53T^{2} $
	59	$ 1 - 2.20iT - 59T^{2} $
	61	$ 1 + 12.4iT - 61T^{2} $
	67	$ 1 + 4iT - 67T^{2} $
	71	$ 1 - 8.41T + 71T^{2} $
	73	$ 1 - 6T + 73T^{2} $
	79	$ 1 + 5.42T + 79T^{2} $
	83	$ 1 - 3.25iT - 83T^{2} $
	89	$ 1 - 13.2T + 89T^{2} $
	97	$ 1 + 4.84T + 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

−7.58730969959535617130633664733, −6.57746317933557400775364514737, −6.44319864428260609074225924516, −5.50274330877248678992803336466, −4.88670187195652454756430569510, −3.64236859473546134590687693704, −3.21567407608056755690778259755, −2.59265841191867027365937453863, −1.01314077577004226059198036795, −0.21058909568908361275071462275, 1.31758176966735746381519597889, 2.28280176566999872321957581879, 3.18193335080929511345923529794, 3.87364920090578492588822979184, 4.71938109121821906539495118283, 5.41917431970932102819793373526, 6.38623506150172537252461130712, 6.80682043377046641937932605855, 7.42997838285985060512158510692, 8.209431612158105317565823693719

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7463609261\)
\(L(\frac12)\)	\(\approx\)	\(0.7463609261\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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