LMFDB - L-function 2-7200-12.11-c1-0-51

Dirichlet series

L(s) = 1

− 0.864i·7^-s − 3.90·11^-s − 1.13·13^-s + 3.71i·17^-s − 1.72i·19^-s + 9.03·23^-s − 1.26i·29^-s − 3.25i·31^-s − 6.38·37^-s + 6.39i·41^-s + 4.77i·43^-s − 4.59·47^-s + 6.25·49^-s − 8.98i·53^-s + 8.50·59^-s + ⋯

L(s) = 1

− 0.326i·7^-s − 1.17·11^-s − 0.314·13^-s + 0.900i·17^-s − 0.396i·19^-s + 1.88·23^-s − 0.235i·29^-s − 0.584i·31^-s − 1.05·37^-s + 0.998i·41^-s + 0.728i·43^-s − 0.670·47^-s + 0.893·49^-s − 1.23i·53^-s + 1.10·59^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.169 + 0.985i$
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:	$(2,\ 7200,\ (\ :1/2),\ -0.169 + 0.985i)$

Particular Values

$L(1)$	$\approx$	$1.093863007$
$L(\frac12)$	$\approx$	$1.093863007$
$L(\frac{3}{2})$		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 + 0.864iT - 7T^{2} $
	11	$ 1 + 3.90T + 11T^{2} $
	13	$ 1 + 1.13T + 13T^{2} $
	17	$ 1 - 3.71iT - 17T^{2} $
	19	$ 1 + 1.72iT - 19T^{2} $
	23	$ 1 - 9.03T + 23T^{2} $
	29	$ 1 + 1.26iT - 29T^{2} $
	31	$ 1 + 3.25iT - 31T^{2} $
	37	$ 1 + 6.38T + 37T^{2} $
	41	$ 1 - 6.39iT - 41T^{2} $
	43	$ 1 - 4.77iT - 43T^{2} $
	47	$ 1 + 4.59T + 47T^{2} $
	53	$ 1 + 8.98iT - 53T^{2} $
	59	$ 1 - 8.50T + 59T^{2} $
	61	$ 1 + 9.04T + 61T^{2} $
	67	$ 1 + 11.0iT - 67T^{2} $
	71	$ 1 - 8.10T + 71T^{2} $
	73	$ 1 + 4.47T + 73T^{2} $
	79	$ 1 - 14.2iT - 79T^{2} $
	83	$ 1 - 8.10T + 83T^{2} $
	89	$ 1 + 3.56iT - 89T^{2} $
	97	$ 1 + 10.9T + 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

−7.80922174565883847952931769566, −7.00582670933271339580866699229, −6.47535387844405639238829532555, −5.44825106470789688381856287627, −5.00120958073113023311031622383, −4.18180243882096917340444277450, −3.22749384328216555661462050537, −2.57125633078204057177721757761, −1.50953266960931074906199370426, −0.30011146232006298849691846164, 0.968147040408770659849810108576, 2.23177294681288996741266411928, 2.88012160000052753337808051830, 3.65060307243604074730522312626, 4.87947289751634454955933479850, 5.14911529443120394690193669108, 5.88806425951177205944033185867, 6.99020156849707454497697531438, 7.26725571452089843310809862264, 8.091731085056496523874799086991

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.093863007\)
\(L(\frac12)\)	\(\approx\)	\(1.093863007\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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