LMFDB - L-function 2-4140-69.68-c1-0-10

Dirichlet series

L(s) = 1

− 5^-s + 4.85i·7^-s + 5.32·11^-s − 0.812·13^-s + 0.626·17^-s + 6.95i·19^-s + (4.49 + 1.66i)23^-s + 25^-s + 2.93i·29^-s + 5.30·31^-s − 4.85i·35^-s − 8.73i·37^-s − 4.04i·41^-s + 2.20i·43^-s + 2.43i·47^-s + ⋯

L(s) = 1

− 0.447·5^-s + 1.83i·7^-s + 1.60·11^-s − 0.225·13^-s + 0.152·17^-s + 1.59i·19^-s + (0.937 + 0.347i)23^-s + 0.200·25^-s + 0.544i·29^-s + 0.952·31^-s − 0.821i·35^-s − 1.43i·37^-s − 0.631i·41^-s + 0.335i·43^-s + 0.355i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4140$ = $2^{2} \cdot 3^{2} \cdot 5 \cdot 23$
Sign:	$-0.258 - 0.966i$
Analytic conductor:	$33.0580$
Root analytic conductor:	$5.74961$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4140} (1241, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4140,\ (\ :1/2),\ -0.258 - 0.966i)$

Particular Values

$L(1)$	$\approx$	$1.830311070$
$L(\frac12)$	$\approx$	$1.830311070$
$L(\frac{3}{2})$		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 + T $
	23	$ 1 + (-4.49 - 1.66i)T $
good	7	$ 1 - 4.85iT - 7T^{2} $
	11	$ 1 - 5.32T + 11T^{2} $
	13	$ 1 + 0.812T + 13T^{2} $
	17	$ 1 - 0.626T + 17T^{2} $
	19	$ 1 - 6.95iT - 19T^{2} $
	29	$ 1 - 2.93iT - 29T^{2} $
	31	$ 1 - 5.30T + 31T^{2} $
	37	$ 1 + 8.73iT - 37T^{2} $
	41	$ 1 + 4.04iT - 41T^{2} $
	43	$ 1 - 2.20iT - 43T^{2} $
	47	$ 1 - 2.43iT - 47T^{2} $
	53	$ 1 - 4.08T + 53T^{2} $
	59	$ 1 - 4.20iT - 59T^{2} $
	61	$ 1 + 12.6iT - 61T^{2} $
	67	$ 1 - 11.8iT - 67T^{2} $
	71	$ 1 - 5.69iT - 71T^{2} $
	73	$ 1 + 7.08T + 73T^{2} $
	79	$ 1 + 11.6iT - 79T^{2} $
	83	$ 1 + 2.07T + 83T^{2} $
	89	$ 1 - 14.9T + 89T^{2} $
	97	$ 1 + 3.75iT - 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

−8.785000427116721758456345935447, −8.017203407062583012192951691431, −7.16265344741714409405213726329, −6.29221462100607727204950402709, −5.76731195347004526814236094601, −4.98490894960358138804130106203, −3.97207953417915909569418005881, −3.25264647463696162677790616094, −2.24270788174574763622379260551, −1.28277627144128937860442293889, 0.60965218417999692610662033277, 1.30356813971927915074803395408, 2.86437375511524844061702350763, 3.70191620340520300413811798021, 4.42951606570275929571362171928, 4.84815359753208642883873901414, 6.35011992544653258296232570808, 6.83872748643613226438148967042, 7.29246317228522077387231106082, 8.152910577424278840063913540580

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.830311070\)
\(L(\frac12)\)	\(\approx\)	\(1.830311070\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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