LMFDB - L-function 2-1200-5.4-c3-0-28

Dirichlet series

L(s) = 1

− 3i·3^-s + i·7^-s − 9·9^-s − 42·11^-s + 67i·13^-s + 54i·17^-s − 115·19^-s + 3·21^-s − 162i·23^-s + 27i·27^-s + 210·29^-s + 193·31^-s + 126i·33^-s − 286i·37^-s + 201·39^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.0539i·7^-s − 0.333·9^-s − 1.15·11^-s + 1.42i·13^-s + 0.770i·17^-s − 1.38·19^-s + 0.0311·21^-s − 1.46i·23^-s + 0.192i·27^-s + 1.34·29^-s + 1.11·31^-s + 0.664i·33^-s − 1.27i·37^-s + 0.825·39^-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

$L(2)$	$\approx$	$1.442047289$
$L(\frac12)$	$\approx$	$1.442047289$
$L(\frac{5}{2})$		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 - iT - 343T^{2} $
	11	$ 1 + 42T + 1.33e3T^{2} $
	13	$ 1 - 67iT - 2.19e3T^{2} $
	17	$ 1 - 54iT - 4.91e3T^{2} $
	19	$ 1 + 115T + 6.85e3T^{2} $
	23	$ 1 + 162iT - 1.21e4T^{2} $
	29	$ 1 - 210T + 2.43e4T^{2} $
	31	$ 1 - 193T + 2.97e4T^{2} $
	37	$ 1 + 286iT - 5.06e4T^{2} $
	41	$ 1 - 12T + 6.89e4T^{2} $
	43	$ 1 - 263iT - 7.95e4T^{2} $
	47	$ 1 + 414iT - 1.03e5T^{2} $
	53	$ 1 - 192iT - 1.48e5T^{2} $
	59	$ 1 - 690T + 2.05e5T^{2} $
	61	$ 1 + 733T + 2.26e5T^{2} $
	67	$ 1 + 299iT - 3.00e5T^{2} $
	71	$ 1 - 228T + 3.57e5T^{2} $
	73	$ 1 + 938iT - 3.89e5T^{2} $
	79	$ 1 + 160T + 4.93e5T^{2} $
	83	$ 1 + 462iT - 5.71e5T^{2} $
	89	$ 1 - 240T + 7.04e5T^{2} $
	97	$ 1 + 511iT - 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

−8.945668163689817061408387429813, −8.463137815332931926072856704930, −7.62798733774925553933358993244, −6.57555185584490663999418252963, −6.18221641425964079490542097212, −4.86019748160008048949820800743, −4.12690351616618969844920618144, −2.64437790840564291877786052782, −1.95611203160081603962564116070, −0.46448602788039276588287141850, 0.77327502140195424191626100072, 2.51198740043673519225312159829, 3.21226214650368216767803415266, 4.45896263066938463554294840083, 5.22462872984904700318983263336, 5.97952787468616285686080369476, 7.12148276795583232838510876792, 8.078744092627659192562006052269, 8.541474982401347227654014230095, 9.761836937519103171611012461788

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.442047289\)
\(L(\frac12)\)	\(\approx\)	\(1.442047289\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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