LMFDB - L-function 2-7200-40.29-c1-0-64

Dirichlet series

L(s) = 1

− 1.97i·7^-s + 1.43i·11^-s + 0.241·13^-s − 7.38i·17^-s − 3.04i·19^-s + 0.874i·23^-s + 9.07i·29^-s + 7.44·31^-s + 8.81·37^-s + 1.91·41^-s − 11.2·43^-s + 3.34i·47^-s + 3.09·49^-s + 9.20·53^-s − 6.43i·59^-s + ⋯

L(s) = 1

− 0.747i·7^-s + 0.431i·11^-s + 0.0669·13^-s − 1.79i·17^-s − 0.697i·19^-s + 0.182i·23^-s + 1.68i·29^-s + 1.33·31^-s + 1.44·37^-s + 0.298·41^-s − 1.71·43^-s + 0.487i·47^-s + 0.441·49^-s + 1.26·53^-s − 0.837i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0439 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0439 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7200$ = $2^{5} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.0439 + 0.999i$
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:	$(2,\ 7200,\ (\ :1/2),\ -0.0439 + 0.999i)$

Particular Values

$L(1)$	$\approx$	$1.674330982$
$L(\frac12)$	$\approx$	$1.674330982$
$L(\frac{3}{2})$		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 + 1.97iT - 7T^{2} $
	11	$ 1 - 1.43iT - 11T^{2} $
	13	$ 1 - 0.241T + 13T^{2} $
	17	$ 1 + 7.38iT - 17T^{2} $
	19	$ 1 + 3.04iT - 19T^{2} $
	23	$ 1 - 0.874iT - 23T^{2} $
	29	$ 1 - 9.07iT - 29T^{2} $
	31	$ 1 - 7.44T + 31T^{2} $
	37	$ 1 - 8.81T + 37T^{2} $
	41	$ 1 - 1.91T + 41T^{2} $
	43	$ 1 + 11.2T + 43T^{2} $
	47	$ 1 - 3.34iT - 47T^{2} $
	53	$ 1 - 9.20T + 53T^{2} $
	59	$ 1 + 6.43iT - 59T^{2} $
	61	$ 1 + 4.57iT - 61T^{2} $
	67	$ 1 + 4.86T + 67T^{2} $
	71	$ 1 + 8.21T + 71T^{2} $
	73	$ 1 + 4.12iT - 73T^{2} $
	79	$ 1 + 13.6T + 79T^{2} $
	83	$ 1 - 12.3T + 83T^{2} $
	89	$ 1 + 8.08T + 89T^{2} $
	97	$ 1 + 10.6iT - 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

−7.57631166150170481980249654073, −7.06374547044650765823589675682, −6.58855033649889228938908080644, −5.53235893279512089983663753720, −4.78604609804442416401909732653, −4.33876275725226853005380951207, −3.21316238706489439217635566331, −2.65061806973662721047348815099, −1.41136188090824498767855737261, −0.45750316198845612219206693297, 1.05918815912529860641636594603, 2.11298363362580211365559204388, 2.84430624785790529383025910874, 3.88168039119754903235032389586, 4.38128569019584054774962758591, 5.51066250106829641881749462335, 6.03076100529465839814556726147, 6.45360459886979223562126177091, 7.55066613516163480594457559350, 8.330940424348490587047165549853

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.674330982\)
\(L(\frac12)\)	\(\approx\)	\(1.674330982\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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