LMFDB - L-function 2-675-5.4-c3-0-10

Dirichlet series

L(s) = 1

+ 5.20i·2^-s − 19.0·4^-s − 24.4i·7^-s − 57.4i·8^-s + 28.9·11^-s − 65.3i·13^-s + 126.·14^-s + 146.·16^-s + 68.1i·17^-s − 104.·19^-s + 150. i·22^-s + 154. i·23^-s + 340.·26^-s + 464. i·28^-s − 205.·29^-s + ⋯

L(s) = 1

+ 1.83i·2^-s − 2.38·4^-s − 1.31i·7^-s − 2.53i·8^-s + 0.794·11^-s − 1.39i·13^-s + 2.42·14^-s + 2.28·16^-s + 0.972i·17^-s − 1.26·19^-s + 1.46i·22^-s + 1.40i·23^-s + 2.56·26^-s + 3.13i·28^-s − 1.31·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$675$ = $3^{3} \cdot 5^{2}$
Sign:	$-0.894 + 0.447i$
Analytic conductor:	$39.8262$
Root analytic conductor:	$6.31080$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{675} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 675,\ (\ :3/2),\ -0.894 + 0.447i)$

Particular Values

$L(2)$	$\approx$	$0.9108728639$
$L(\frac12)$	$\approx$	$0.9108728639$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 $
	5	$ 1 $
good	2	$ 1 - 5.20iT - 8T^{2} $
	7	$ 1 + 24.4iT - 343T^{2} $
	11	$ 1 - 28.9T + 1.33e3T^{2} $
	13	$ 1 + 65.3iT - 2.19e3T^{2} $
	17	$ 1 - 68.1iT - 4.91e3T^{2} $
	19	$ 1 + 104.T + 6.85e3T^{2} $
	23	$ 1 - 154. iT - 1.21e4T^{2} $
	29	$ 1 + 205.T + 2.43e4T^{2} $
	31	$ 1 + 18.2T + 2.97e4T^{2} $
	37	$ 1 - 337. iT - 5.06e4T^{2} $
	41	$ 1 - 195.T + 6.89e4T^{2} $
	43	$ 1 - 334. iT - 7.95e4T^{2} $
	47	$ 1 + 5.00iT - 1.03e5T^{2} $
	53	$ 1 - 319. iT - 1.48e5T^{2} $
	59	$ 1 - 430.T + 2.05e5T^{2} $
	61	$ 1 - 594.T + 2.26e5T^{2} $
	67	$ 1 + 195. iT - 3.00e5T^{2} $
	71	$ 1 + 425.T + 3.57e5T^{2} $
	73	$ 1 - 929. iT - 3.89e5T^{2} $
	79	$ 1 + 24.4T + 4.93e5T^{2} $
	83	$ 1 - 545. iT - 5.71e5T^{2} $
	89	$ 1 + 84.1T + 7.04e5T^{2} $
	97	$ 1 + 827. 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

−10.28371233728134914177747165466, −9.528880245748807785994076805289, −8.452119729075415373182926261486, −7.82684274758891619733028811131, −7.09919984975649201999954642203, −6.26206735451915188783228606403, −5.48682137259576932616183518011, −4.30179940037128473431029182355, −3.64927030376496375798406338730, −1.08852767376218384787376887657, 0.29955255620455492871205772705, 1.95236711675211216672998741928, 2.40269303659349924327912697970, 3.79709724374221491828826995917, 4.54741818266149845123578345098, 5.68134347864552273247726934150, 6.88478104816109215733389150329, 8.575319851673987458644865563411, 9.035764448456451603513606681891, 9.558316772029014002458158399446

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.9108728639\)
\(L(\frac12)\)	\(\approx\)	\(0.9108728639\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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