LMFDB - L-function 2-4140-345.344-c1-0-47

Dirichlet series

L(s) = 1

− 2.23i·5^-s − 4.05·7^-s − 8.20i·17^-s − 4.79i·23^-s − 5.00·25^-s − 3.74i·29^-s − 7.95·31^-s + 9.06i·35^-s + 12.0·37^-s + 12.1i·41^-s − 0.457·43^-s + 9.43·49^-s + 11.3i·53^-s + 7.84i·59^-s − 13.8·67^-s + ⋯

L(s) = 1

− 0.999i·5^-s − 1.53·7^-s − 1.99i·17^-s − 0.999i·23^-s − 1.00·25^-s − 0.694i·29^-s − 1.42·31^-s + 1.53i·35^-s + 1.97·37^-s + 1.90i·41^-s − 0.0698·43^-s + 1.34·49^-s + 1.55i·53^-s + 1.02i·59^-s − 1.69·67^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4140$ = $2^{2} \cdot 3^{2} \cdot 5 \cdot 23$
Sign:	$-0.577 - 0.816i$
Analytic conductor:	$33.0580$
Root analytic conductor:	$5.74961$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4140} (2069, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4140,\ (\ :1/2),\ -0.577 - 0.816i)$

Particular Values

$L(1)$	$\approx$	$0.1292805765$
$L(\frac12)$	$\approx$	$0.1292805765$
$L(\frac{3}{2})$		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 + 2.23iT $
	23	$ 1 + 4.79iT $
good	7	$ 1 + 4.05T + 7T^{2} $
	11	$ 1 + 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 + 8.20iT - 17T^{2} $
	19	$ 1 - 19T^{2} $
	29	$ 1 + 3.74iT - 29T^{2} $
	31	$ 1 + 7.95T + 31T^{2} $
	37	$ 1 - 12.0T + 37T^{2} $
	41	$ 1 - 12.1iT - 41T^{2} $
	43	$ 1 + 0.457T + 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 11.3iT - 53T^{2} $
	59	$ 1 - 7.84iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 + 13.8T + 67T^{2} $
	71	$ 1 + 15.3iT - 71T^{2} $
	73	$ 1 - 73T^{2} $
	79	$ 1 - 79T^{2} $
	83	$ 1 + 6.86iT - 83T^{2} $
	89	$ 1 + 89T^{2} $
	97	$ 1 - 10.4T + 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

−7.84641003323985735315018773938, −7.29863761109831444915236477009, −6.32518744471331978776872124251, −5.86035166999181336803568642395, −4.82161725967395322814641128142, −4.26748077511519614948912581187, −3.14470434907458690427775833315, −2.50230254478389214526631702887, −0.974523503857372395258663994797, −0.04211971676956246110680753779, 1.72734535335252377659308421877, 2.73635739374214025993851747760, 3.66526302329902071071878233936, 3.88445781182969139075518772861, 5.50733881308693157773234012667, 6.02449845691066815773780363576, 6.69462616002408393342879215043, 7.27778537254966168453935458803, 8.081620091825859009057183482231, 9.007398422335593155421181488943

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.1292805765\)
\(L(\frac12)\)	\(\approx\)	\(0.1292805765\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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