LMFDB - L-function 2-600-5.4-c3-0-8

Dirichlet series

L(s) = 1

− 3i·3^-s + 5i·7^-s − 9·9^-s + 14·11^-s − i·13^-s + 46i·17^-s − 19·19^-s + 15·21^-s + 46i·23^-s + 27i·27^-s − 14·29^-s + 133·31^-s − 42i·33^-s + 258i·37^-s − 3·39^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.269i·7^-s − 0.333·9^-s + 0.383·11^-s − 0.0213i·13^-s + 0.656i·17^-s − 0.229·19^-s + 0.155·21^-s + 0.417i·23^-s + 0.192i·27^-s − 0.0896·29^-s + 0.770·31^-s − 0.221i·33^-s + 1.14i·37^-s − 0.0123·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$600$ = $2^{3} \cdot 3 \cdot 5^{2}$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$35.4011$
Root analytic conductor:	$5.94988$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{600} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 600,\ (\ :3/2),\ 0.894 - 0.447i)$

Particular Values

$L(2)$	$\approx$	$1.772316565$
$L(\frac12)$	$\approx$	$1.772316565$
$L(\frac{5}{2})$		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 + 3iT $
	5	$ 1 $
good	7	$ 1 - 5iT - 343T^{2} $
	11	$ 1 - 14T + 1.33e3T^{2} $
	13	$ 1 + iT - 2.19e3T^{2} $
	17	$ 1 - 46iT - 4.91e3T^{2} $
	19	$ 1 + 19T + 6.85e3T^{2} $
	23	$ 1 - 46iT - 1.21e4T^{2} $
	29	$ 1 + 14T + 2.43e4T^{2} $
	31	$ 1 - 133T + 2.97e4T^{2} $
	37	$ 1 - 258iT - 5.06e4T^{2} $
	41	$ 1 - 84T + 6.89e4T^{2} $
	43	$ 1 - 167iT - 7.95e4T^{2} $
	47	$ 1 - 410iT - 1.03e5T^{2} $
	53	$ 1 + 456iT - 1.48e5T^{2} $
	59	$ 1 - 194T + 2.05e5T^{2} $
	61	$ 1 + 17T + 2.26e5T^{2} $
	67	$ 1 - 653iT - 3.00e5T^{2} $
	71	$ 1 - 828T + 3.57e5T^{2} $
	73	$ 1 + 570iT - 3.89e5T^{2} $
	79	$ 1 - 552T + 4.93e5T^{2} $
	83	$ 1 + 142iT - 5.71e5T^{2} $
	89	$ 1 - 1.10e3T + 7.04e5T^{2} $
	97	$ 1 - 841iT - 9.12e5T^{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

−10.33632274258952882546110309110, −9.388110656755244807452141179795, −8.478318440346804837615389793335, −7.73148481151146225187159570789, −6.64919088844804273824691245604, −5.96241722043928980122202333366, −4.78930129485965841094195935731, −3.55071016051753500917026201096, −2.29904763295894959878539969108, −1.06905548052865161491382716048, 0.61610103676195062383139678494, 2.32168395748992946324983042550, 3.60669195043565804127643780183, 4.50809847561764285088880805692, 5.52111305824844172373223341601, 6.58902989762741280644121028035, 7.52872087305653300171032780579, 8.612421736483666771433280564708, 9.342480210181074280669813404427, 10.22257743998145646090133329610

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.772316565\)
\(L(\frac12)\)	\(\approx\)	\(1.772316565\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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