LMFDB - L-function 2-7200-8.5-c1-0-64

Dirichlet series

L(s) = 1

− 1.33·7^-s + 2.94i·11^-s − 2.04i·13^-s − 3.61·17^-s − 5.35i·19^-s + 8.59·23^-s − 5.26i·29^-s + 2.08·31^-s + 6.55i·37^-s − 7.02·41^-s + 8.50i·43^-s + 9.97·47^-s − 5.22·49^-s + 6.12i·53^-s + 4.75i·59^-s + ⋯

L(s) = 1

− 0.504·7^-s + 0.887i·11^-s − 0.566i·13^-s − 0.876·17^-s − 1.22i·19^-s + 1.79·23^-s − 0.977i·29^-s + 0.373·31^-s + 1.07i·37^-s − 1.09·41^-s + 1.29i·43^-s + 1.45·47^-s − 0.745·49^-s + 0.841i·53^-s + 0.618i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0418 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0418 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.0418 + 0.999i$
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:	$(2,\ 7200,\ (\ :1/2),\ -0.0418 + 0.999i)$

Particular Values

$L(1)$	$\approx$	$1.157027435$
$L(\frac12)$	$\approx$	$1.157027435$
$L(\frac{3}{2})$		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 + 1.33T + 7T^{2} $
	11	$ 1 - 2.94iT - 11T^{2} $
	13	$ 1 + 2.04iT - 13T^{2} $
	17	$ 1 + 3.61T + 17T^{2} $
	19	$ 1 + 5.35iT - 19T^{2} $
	23	$ 1 - 8.59T + 23T^{2} $
	29	$ 1 + 5.26iT - 29T^{2} $
	31	$ 1 - 2.08T + 31T^{2} $
	37	$ 1 - 6.55iT - 37T^{2} $
	41	$ 1 + 7.02T + 41T^{2} $
	43	$ 1 - 8.50iT - 43T^{2} $
	47	$ 1 - 9.97T + 47T^{2} $
	53	$ 1 - 6.12iT - 53T^{2} $
	59	$ 1 - 4.75iT - 59T^{2} $
	61	$ 1 + 8.51iT - 61T^{2} $
	67	$ 1 + 10.6iT - 67T^{2} $
	71	$ 1 + 2.62T + 71T^{2} $
	73	$ 1 + 15.3T + 73T^{2} $
	79	$ 1 + 10.4T + 79T^{2} $
	83	$ 1 - 1.52iT - 83T^{2} $
	89	$ 1 - 12.7T + 89T^{2} $
	97	$ 1 + 13.4T + 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

−7.65603467687674617803436462468, −6.93763102770457330486503020471, −6.55369749734176221620096691524, −5.63889677872251459421447811207, −4.70453548562190351068299931737, −4.42170188017653978586981017883, −3.07699804230215937638578903956, −2.71835642037563760169750496287, −1.51839862921497071923731314970, −0.31901497607796968337153564497, 0.978347558055241364444414408545, 2.05200140953492433358045762877, 3.05481923262183232479697529129, 3.67014134243147689296407278139, 4.49800763056425378123960216995, 5.41455016960889793102254602476, 5.95350223041039458257479791307, 6.89574618131631224030349079246, 7.12927493945806594969320076055, 8.248611178761368979952338130738

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.157027435\)
\(L(\frac12)\)	\(\approx\)	\(1.157027435\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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