LMFDB - L-function 2-3200-8.5-c1-0-48

Dirichlet series

L(s) = 1

+ 3·9^-s − 4i·13^-s + 2·17^-s + 4i·29^-s − 12i·37^-s + 10·41^-s − 7·49^-s + 4i·53^-s − 12i·61^-s − 6·73^-s + 9·81^-s − 10·89^-s + 18·97^-s − 20i·101^-s + 20i·109^-s + ⋯

L(s) = 1

+ 9^-s − 1.10i·13^-s + 0.485·17^-s + 0.742i·29^-s − 1.97i·37^-s + 1.56·41^-s − 49^-s + 0.549i·53^-s − 1.53i·61^-s − 0.702·73^-s + 81^-s − 1.05·89^-s + 1.82·97^-s − 1.99i·101^-s + 1.91i·109^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3200$ = $2^{7} \cdot 5^{2}$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$25.5521$
Root analytic conductor:	$5.05491$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3200} (1601, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3200,\ (\ :1/2),\ 0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.972103549$
$L(\frac12)$	$\approx$	$1.972103549$
$L(\frac{3}{2})$		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 - 3T^{2} $
	7	$ 1 + 7T^{2} $
	11	$ 1 - 11T^{2} $
	13	$ 1 + 4iT - 13T^{2} $
	17	$ 1 - 2T + 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 - 4iT - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 12iT - 37T^{2} $
	41	$ 1 - 10T + 41T^{2} $
	43	$ 1 - 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 4iT - 53T^{2} $
	59	$ 1 - 59T^{2} $
	61	$ 1 + 12iT - 61T^{2} $
	67	$ 1 - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 6T + 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 - 83T^{2} $
	89	$ 1 + 10T + 89T^{2} $
	97	$ 1 - 18T + 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

−8.553248814621543414559476563048, −7.62990945226927632845305968034, −7.32439499380261798354075326981, −6.26233683457792968630773813863, −5.54393653160422277646327294623, −4.71942789328790503606459580393, −3.84037559500478732364104226705, −3.00619302827288649471824217852, −1.85469028369450052785797693339, −0.70883904967130064489406282429, 1.11944134244234932461499603278, 2.07621330713333084387651272421, 3.21659745691824248076399696267, 4.24092513712175700102627758335, 4.69748299905297721054346964029, 5.82520886959755525616969532327, 6.57804545545840312001509132333, 7.23893537670033320295523467281, 7.962748886379707596434734267393, 8.771621238555295582313835371606

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.972103549\)
\(L(\frac12)\)	\(\approx\)	\(1.972103549\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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