LMFDB - L-function 2-3200-5.4-c1-0-27

Dirichlet series

L(s) = 1

− 2.41i·3^-s + 0.828i·7^-s − 2.82·9^-s − 5.24·11^-s + 5.65i·13^-s + 0.171i·17^-s + 1.58·19^-s + 1.99·21^-s + 4.82i·23^-s − 0.414i·27^-s + 8·29^-s + 0.828·31^-s + 12.6i·33^-s − 7.65i·37^-s + 13.6·39^-s + ⋯

L(s) = 1

− 1.39i·3^-s + 0.313i·7^-s − 0.942·9^-s − 1.58·11^-s + 1.56i·13^-s + 0.0416i·17^-s + 0.363·19^-s + 0.436·21^-s + 1.00i·23^-s − 0.0797i·27^-s + 1.48·29^-s + 0.148·31^-s + 2.20i·33^-s − 1.25i·37^-s + 2.18·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3200$ = $2^{7} \cdot 5^{2}$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$25.5521$
Root analytic conductor:	$5.05491$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3200} (2049, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3200,\ (\ :1/2),\ 0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$1.573561096$
$L(\frac12)$	$\approx$	$1.573561096$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
good	3	$ 1 + 2.41iT - 3T^{2} $
	7	$ 1 - 0.828iT - 7T^{2} $
	11	$ 1 + 5.24T + 11T^{2} $
	13	$ 1 - 5.65iT - 13T^{2} $
	17	$ 1 - 0.171iT - 17T^{2} $
	19	$ 1 - 1.58T + 19T^{2} $
	23	$ 1 - 4.82iT - 23T^{2} $
	29	$ 1 - 8T + 29T^{2} $
	31	$ 1 - 0.828T + 31T^{2} $
	37	$ 1 + 7.65iT - 37T^{2} $
	41	$ 1 - 10.6T + 41T^{2} $
	43	$ 1 + 10iT - 43T^{2} $
	47	$ 1 + 9.65iT - 47T^{2} $
	53	$ 1 + 7.65iT - 53T^{2} $
	59	$ 1 + 3.65T + 59T^{2} $
	61	$ 1 + 6T + 61T^{2} $
	67	$ 1 + 2.75iT - 67T^{2} $
	71	$ 1 - 9.65T + 71T^{2} $
	73	$ 1 - 5.82iT - 73T^{2} $
	79	$ 1 - 12.8T + 79T^{2} $
	83	$ 1 + 1.24iT - 83T^{2} $
	89	$ 1 + 6.17T + 89T^{2} $
	97	$ 1 - 17.3iT - 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

−8.399205988583068362521587816140, −7.59438604661351827332896620175, −7.19283281870393667078837072609, −6.40118596675179491683494071484, −5.63087714976238310658672095417, −4.85755038359512127093859230172, −3.70228757176455576848050922442, −2.43362422814152774647991266828, −2.02698896702243112711522493667, −0.70762900335616452250646992932, 0.77955967243387585716311151967, 2.87260710518587221396619414197, 2.97135219300863248273948531392, 4.37796570950588149668113930921, 4.78945963628878229836538094404, 5.56132818329605838215511769342, 6.32595186564942816715541268597, 7.66916444501208227545627127408, 7.980328954507207595381959617025, 8.864397634245682150303722798086

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.573561096\)
\(L(\frac12)\)	\(\approx\)	\(1.573561096\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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