LMFDB - L-function 2-1200-5.4-c3-0-6

Dirichlet series

L(s) = 1

− 3i·3^-s − 19i·7^-s − 9·9^-s − 22·11^-s + i·13^-s + 58i·17^-s − 53·19^-s − 57·21^-s − 58i·23^-s + 27i·27^-s − 22·29^-s + 35·31^-s + 66i·33^-s + 270i·37^-s + 3·39^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 1.02i·7^-s − 0.333·9^-s − 0.603·11^-s + 0.0213i·13^-s + 0.827i·17^-s − 0.639·19^-s − 0.592·21^-s − 0.525i·23^-s + 0.192i·27^-s − 0.140·29^-s + 0.202·31^-s + 0.348i·33^-s + 1.19i·37^-s + 0.0123·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1200$ = $2^{4} \cdot 3 \cdot 5^{2}$
Sign:	$0.447 - 0.894i$
Analytic conductor:	$70.8022$
Root analytic conductor:	$8.41440$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1200} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1200,\ (\ :3/2),\ 0.447 - 0.894i)$

Particular Values

$L(2)$	$\approx$	$0.8559384997$
$L(\frac12)$	$\approx$	$0.8559384997$
$L(\frac{5}{2})$		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 + 3iT $
	5	$ 1 $
good	7	$ 1 + 19iT - 343T^{2} $
	11	$ 1 + 22T + 1.33e3T^{2} $
	13	$ 1 - iT - 2.19e3T^{2} $
	17	$ 1 - 58iT - 4.91e3T^{2} $
	19	$ 1 + 53T + 6.85e3T^{2} $
	23	$ 1 + 58iT - 1.21e4T^{2} $
	29	$ 1 + 22T + 2.43e4T^{2} $
	31	$ 1 - 35T + 2.97e4T^{2} $
	37	$ 1 - 270iT - 5.06e4T^{2} $
	41	$ 1 + 468T + 6.89e4T^{2} $
	43	$ 1 - 431iT - 7.95e4T^{2} $
	47	$ 1 + 230iT - 1.03e5T^{2} $
	53	$ 1 - 1.48e5T^{2} $
	59	$ 1 - 446T + 2.05e5T^{2} $
	61	$ 1 - 127T + 2.26e5T^{2} $
	67	$ 1 + 811iT - 3.00e5T^{2} $
	71	$ 1 + 36T + 3.57e5T^{2} $
	73	$ 1 - 522iT - 3.89e5T^{2} $
	79	$ 1 - 1.36e3T + 4.93e5T^{2} $
	83	$ 1 - 1.13e3iT - 5.71e5T^{2} $
	89	$ 1 + 144T + 7.04e5T^{2} $
	97	$ 1 - 1.07e3iT - 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

−9.623107330551412169859254668270, −8.325992981381608068451800952341, −8.055417334497381258326141441641, −6.91395659246519763181270636426, −6.45058783078453218718508372348, −5.28208727070345760305888995764, −4.31584814311258828896028794723, −3.31765909873814050097685242656, −2.10737766527474019519294205109, −0.979405449929068767841444038584, 0.22592613429839327968996744287, 2.03418719199762762794610146212, 2.91694161138341732305412082343, 3.99140701330680510945625799490, 5.14567626251052141538850971269, 5.58368938233519999418879098240, 6.68906230356035864583033231635, 7.68271636876656291122632637394, 8.631818648224490421994522047989, 9.146186763236154565473307030730

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.8559384997\)
\(L(\frac12)\)	\(\approx\)	\(0.8559384997\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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