LMFDB - L-function 2-7200-8.5-c1-0-76

Dirichlet series

L(s) = 1

− 0.746·7^-s − 5.36i·11^-s − 2.92i·13^-s − 2.13·17^-s + 1.73i·19^-s + 7.49·23^-s + 6.74i·29^-s − 2.64·31^-s + 1.07i·37^-s + 11.2·41^-s − 7.44i·43^-s + 1.73·47^-s − 6.44·49^-s − 7.72i·53^-s − 6.85i·59^-s + ⋯

L(s) = 1

− 0.282·7^-s − 1.61i·11^-s − 0.811i·13^-s − 0.517·17^-s + 0.397i·19^-s + 1.56·23^-s + 1.25i·29^-s − 0.475·31^-s + 0.176i·37^-s + 1.76·41^-s − 1.13i·43^-s + 0.252·47^-s − 0.920·49^-s − 1.06i·53^-s − 0.892i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.570 + 0.821i$
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:	$(2,\ 7200,\ (\ :1/2),\ -0.570 + 0.821i)$

Particular Values

$L(1)$	$\approx$	$1.289863775$
$L(\frac12)$	$\approx$	$1.289863775$
$L(\frac{3}{2})$		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 + 0.746T + 7T^{2} $
	11	$ 1 + 5.36iT - 11T^{2} $
	13	$ 1 + 2.92iT - 13T^{2} $
	17	$ 1 + 2.13T + 17T^{2} $
	19	$ 1 - 1.73iT - 19T^{2} $
	23	$ 1 - 7.49T + 23T^{2} $
	29	$ 1 - 6.74iT - 29T^{2} $
	31	$ 1 + 2.64T + 31T^{2} $
	37	$ 1 - 1.07iT - 37T^{2} $
	41	$ 1 - 11.2T + 41T^{2} $
	43	$ 1 + 7.44iT - 43T^{2} $
	47	$ 1 - 1.73T + 47T^{2} $
	53	$ 1 + 7.72iT - 53T^{2} $
	59	$ 1 + 6.85iT - 59T^{2} $
	61	$ 1 - 6.45iT - 61T^{2} $
	67	$ 1 + 7.44iT - 67T^{2} $
	71	$ 1 - 13.2T + 71T^{2} $
	73	$ 1 + 0.690T + 73T^{2} $
	79	$ 1 + 2.64T + 79T^{2} $
	83	$ 1 + 5.85iT - 83T^{2} $
	89	$ 1 + 7.59T + 89T^{2} $
	97	$ 1 + 14.1T + 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

−7.73978792698585052085630519262, −6.91860344752422992928499470706, −6.30696580776706757620942082583, −5.49932683226669674233301709694, −5.07478071819534080019698884954, −3.87259096202207704081528193112, −3.27153723774878023055857642717, −2.61178443079405766396475769204, −1.28107548182173767529287796287, −0.34009208820521448309798915286, 1.17135799776190699960105504203, 2.21901890578716118817018658194, 2.82878584796879781353210648996, 4.11295537292324954492590308942, 4.46031689930942014087215149247, 5.25972250067253205297094072640, 6.20591073539357464612241082660, 6.88774581433312803012400289160, 7.30846979748259715301754092201, 8.073641119834073909957572566030

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.289863775\)
\(L(\frac12)\)	\(\approx\)	\(1.289863775\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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