LMFDB - L-function 2-2952-41.40-c1-0-39

Dirichlet series

L(s) = 1

+ 1.60·5^-s + 0.181i·7^-s − 2.05i·11^-s + 0.949i·13^-s − 3.83i·17^-s − 6.15i·19^-s + 4.73·23^-s − 2.42·25^-s − 3.26i·29^-s − 7.12·31^-s + 0.291i·35^-s − 4.42·37^-s + (0.372 + 6.39i)41^-s − 7.54·43^-s − 10.2i·47^-s + ⋯

L(s) = 1

+ 0.717·5^-s + 0.0687i·7^-s − 0.618i·11^-s + 0.263i·13^-s − 0.930i·17^-s − 1.41i·19^-s + 0.987·23^-s − 0.484·25^-s − 0.605i·29^-s − 1.28·31^-s + 0.0493i·35^-s − 0.727·37^-s + (0.0582 + 0.998i)41^-s − 1.15·43^-s − 1.49i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0582 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0582 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2952$ = $2^{3} \cdot 3^{2} \cdot 41$
Sign:	$0.0582 + 0.998i$
Analytic conductor:	$23.5718$
Root analytic conductor:	$4.85508$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2952} (2377, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2952,\ (\ :1/2),\ 0.0582 + 0.998i)$

Particular Values

$L(1)$	$\approx$	$1.689830502$
$L(\frac12)$	$\approx$	$1.689830502$
$L(\frac{3}{2})$		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 $
	41	$ 1 + (-0.372 - 6.39i)T $
good	5	$ 1 - 1.60T + 5T^{2} $
	7	$ 1 - 0.181iT - 7T^{2} $
	11	$ 1 + 2.05iT - 11T^{2} $
	13	$ 1 - 0.949iT - 13T^{2} $
	17	$ 1 + 3.83iT - 17T^{2} $
	19	$ 1 + 6.15iT - 19T^{2} $
	23	$ 1 - 4.73T + 23T^{2} $
	29	$ 1 + 3.26iT - 29T^{2} $
	31	$ 1 + 7.12T + 31T^{2} $
	37	$ 1 + 4.42T + 37T^{2} $
	43	$ 1 + 7.54T + 43T^{2} $
	47	$ 1 + 10.2iT - 47T^{2} $
	53	$ 1 + 0.554iT - 53T^{2} $
	59	$ 1 - 11.6T + 59T^{2} $
	61	$ 1 + 9.80T + 61T^{2} $
	67	$ 1 + 7.91iT - 67T^{2} $
	71	$ 1 + 3.73iT - 71T^{2} $
	73	$ 1 - 12.1T + 73T^{2} $
	79	$ 1 + 8.84iT - 79T^{2} $
	83	$ 1 - 1.70T + 83T^{2} $
	89	$ 1 + 4.76iT - 89T^{2} $
	97	$ 1 - 9.39iT - 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

−8.850463946504120442320849357879, −7.76628801802387251352102866586, −6.95953422222405964965002127268, −6.36977612389958824043747592158, −5.35658736002669903451218926146, −4.92911506209472018646943843468, −3.69990777061748047402783835077, −2.79364610578420736463928388320, −1.89260644021547095338657760799, −0.52066281085968050218160781695, 1.40268589954349397974121017253, 2.14981895577315241833913710910, 3.39042608811034907437630557455, 4.14137429206870761199712459216, 5.30763948230355957160423278126, 5.73829288538697141505940267409, 6.67613592817155349022040418563, 7.38821520147804814871590982405, 8.207073894096243505073139921294, 8.968720457608307240040229669649

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.689830502\)
\(L(\frac12)\)	\(\approx\)	\(1.689830502\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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