LMFDB - L-function 2-2268-21.20-c3-0-13

Dirichlet series

L(s) = 1

− 10.9·5^-s + (11.4 + 14.5i)7^-s + 15.9i·11^-s + 89.4i·13^-s − 106.·17^-s − 8.31i·19^-s + 143. i·23^-s − 4.13·25^-s + 149. i·29^-s + 42.9i·31^-s + (−126. − 159. i)35^-s + 390.·37^-s − 344.·41^-s − 57.1·43^-s − 16.2·47^-s + ⋯

L(s) = 1

− 0.983·5^-s + (0.619 + 0.785i)7^-s + 0.437i·11^-s + 1.90i·13^-s − 1.52·17^-s − 0.100i·19^-s + 1.29i·23^-s − 0.0330·25^-s + 0.959i·29^-s + 0.248i·31^-s + (−0.609 − 0.772i)35^-s + 1.73·37^-s − 1.31·41^-s − 0.202·43^-s − 0.0504·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 + 0.619i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.785 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2268$ = $2^{2} \cdot 3^{4} \cdot 7$
Sign:	$-0.785 + 0.619i$
Analytic conductor:	$133.816$
Root analytic conductor:	$11.5679$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2268} (1133, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2268,\ (\ :3/2),\ -0.785 + 0.619i)$

Particular Values

$L(2)$	$\approx$	$0.7426934548$
$L(\frac12)$	$\approx$	$0.7426934548$
$L(\frac{5}{2})$		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 $
	7	$ 1 + (-11.4 - 14.5i)T $
good	5	$ 1 + 10.9T + 125T^{2} $
	11	$ 1 - 15.9iT - 1.33e3T^{2} $
	13	$ 1 - 89.4iT - 2.19e3T^{2} $
	17	$ 1 + 106.T + 4.91e3T^{2} $
	19	$ 1 + 8.31iT - 6.85e3T^{2} $
	23	$ 1 - 143. iT - 1.21e4T^{2} $
	29	$ 1 - 149. iT - 2.43e4T^{2} $
	31	$ 1 - 42.9iT - 2.97e4T^{2} $
	37	$ 1 - 390.T + 5.06e4T^{2} $
	41	$ 1 + 344.T + 6.89e4T^{2} $
	43	$ 1 + 57.1T + 7.95e4T^{2} $
	47	$ 1 + 16.2T + 1.03e5T^{2} $
	53	$ 1 - 445. iT - 1.48e5T^{2} $
	59	$ 1 + 386.T + 2.05e5T^{2} $
	61	$ 1 + 485. iT - 2.26e5T^{2} $
	67	$ 1 - 503.T + 3.00e5T^{2} $
	71	$ 1 + 751. iT - 3.57e5T^{2} $
	73	$ 1 - 507. iT - 3.89e5T^{2} $
	79	$ 1 + 763.T + 4.93e5T^{2} $
	83	$ 1 - 1.21e3T + 5.71e5T^{2} $
	89	$ 1 - 425.T + 7.04e5T^{2} $
	97	$ 1 - 571. iT - 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

−9.105469996992295660388300461021, −8.420838647177568048535518640519, −7.58713738248783031007950825208, −6.90104760398467332481870877954, −6.13576546931397215991497693633, −4.88613625118946005576859401223, −4.45849040027170637401560103541, −3.55290680659432336733017426442, −2.27220834415686951034667501160, −1.54436816593846658440093725185, 0.19723941260560671550520073272, 0.76328105931784460970149350769, 2.30077560146792666021404127208, 3.32406706849592506152593623336, 4.20009214085021674635355011635, 4.78814039085114310174114708250, 5.86680831461487307999725663282, 6.73967446169443264047147828882, 7.64920694677809796940910615663, 8.152174632380453653277263624139

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.7426934548\)
\(L(\frac12)\)	\(\approx\)	\(0.7426934548\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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