LMFDB - L-function 2-60e2-5.4-c1-0-22

Dirichlet series

L(s) = 1

− 3i·7^-s + 2·11^-s + i·13^-s + 2i·17^-s − 5·19^-s + 6i·23^-s + 10·29^-s + 3·31^-s − 2i·37^-s + 8·41^-s − i·43^-s − 2i·47^-s − 2·49^-s + 4i·53^-s + 10·59^-s + ⋯

L(s) = 1

− 1.13i·7^-s + 0.603·11^-s + 0.277i·13^-s + 0.485i·17^-s − 1.14·19^-s + 1.25i·23^-s + 1.85·29^-s + 0.538·31^-s − 0.328i·37^-s + 1.24·41^-s − 0.152i·43^-s − 0.291i·47^-s − 0.285·49^-s + 0.549i·53^-s + 1.30·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3600$ = $2^{4} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$28.7461$
Root analytic conductor:	$5.36154$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3600} (2449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3600,\ (\ :1/2),\ 0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$1.906328761$
$L(\frac12)$	$\approx$	$1.906328761$
$L(\frac{3}{2})$		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 + 3iT - 7T^{2} $
	11	$ 1 - 2T + 11T^{2} $
	13	$ 1 - iT - 13T^{2} $
	17	$ 1 - 2iT - 17T^{2} $
	19	$ 1 + 5T + 19T^{2} $
	23	$ 1 - 6iT - 23T^{2} $
	29	$ 1 - 10T + 29T^{2} $
	31	$ 1 - 3T + 31T^{2} $
	37	$ 1 + 2iT - 37T^{2} $
	41	$ 1 - 8T + 41T^{2} $
	43	$ 1 + iT - 43T^{2} $
	47	$ 1 + 2iT - 47T^{2} $
	53	$ 1 - 4iT - 53T^{2} $
	59	$ 1 - 10T + 59T^{2} $
	61	$ 1 - 7T + 61T^{2} $
	67	$ 1 + 3iT - 67T^{2} $
	71	$ 1 + 8T + 71T^{2} $
	73	$ 1 + 14iT - 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 - 6iT - 83T^{2} $
	89	$ 1 + 89T^{2} $
	97	$ 1 + 17iT - 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

−8.457189917590419148981404775732, −7.75728981447324310512869143578, −6.93293316396133459635276734796, −6.44051054470360246201953084979, −5.54197284861683970724801721398, −4.34905375740304426911335785130, −4.09179695764320897983563816891, −3.01440658992017247458849886765, −1.80180902406788749030214493980, −0.77438724675417013692147743685, 0.883639595464901708145119393493, 2.32283747100326917277925455106, 2.80907327913162999350199238449, 4.09325649730811148522801939385, 4.76930070963763838901387465898, 5.67703057570696635477832237119, 6.41229401047564575705178209281, 6.91688036260201372110304281150, 8.190273255472999823568622119857, 8.510207643854788456513406531326

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.906328761\)
\(L(\frac12)\)	\(\approx\)	\(1.906328761\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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