LMFDB - L-function 2-3168-44.43-c1-0-53

Dirichlet series

L(s) = 1

+ 3.91·5^-s + 3.25·7^-s + (1.62 + 2.89i)11^-s − 0.831i·13^-s − 6.58i·17^-s − 1.44·19^-s − 0.691i·23^-s + 10.3·25^-s − 6.15i·29^-s − 6.70i·31^-s + 12.7·35^-s − 6.70·37^-s + 3.32i·41^-s + 9.62·43^-s − 2.73i·47^-s + ⋯

L(s) = 1

+ 1.74·5^-s + 1.23·7^-s + (0.490 + 0.871i)11^-s − 0.230i·13^-s − 1.59i·17^-s − 0.331·19^-s − 0.144i·23^-s + 2.06·25^-s − 1.14i·29^-s − 1.20i·31^-s + 2.15·35^-s − 1.10·37^-s + 0.519i·41^-s + 1.46·43^-s − 0.399i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3168$ = $2^{5} \cdot 3^{2} \cdot 11$
Sign:	$0.963 + 0.269i$
Analytic conductor:	$25.2966$
Root analytic conductor:	$5.02957$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3168} (703, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3168,\ (\ :1/2),\ 0.963 + 0.269i)$

Particular Values

$L(1)$	$\approx$	$3.215947671$
$L(\frac12)$	$\approx$	$3.215947671$
$L(\frac{3}{2})$		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 $
	3	$ 1 $
	11	$ 1 + (-1.62 - 2.89i)T $
good	5	$ 1 - 3.91T + 5T^{2} $
	7	$ 1 - 3.25T + 7T^{2} $
	13	$ 1 + 0.831iT - 13T^{2} $
	17	$ 1 + 6.58iT - 17T^{2} $
	19	$ 1 + 1.44T + 19T^{2} $
	23	$ 1 + 0.691iT - 23T^{2} $
	29	$ 1 + 6.15iT - 29T^{2} $
	31	$ 1 + 6.70iT - 31T^{2} $
	37	$ 1 + 6.70T + 37T^{2} $
	41	$ 1 - 3.32iT - 41T^{2} $
	43	$ 1 - 9.62T + 43T^{2} $
	47	$ 1 + 2.73iT - 47T^{2} $
	53	$ 1 + 1.56T + 53T^{2} $
	59	$ 1 - 3.42iT - 59T^{2} $
	61	$ 1 + 11.8iT - 61T^{2} $
	67	$ 1 - 11.9iT - 67T^{2} $
	71	$ 1 - 10.5iT - 71T^{2} $
	73	$ 1 + 3.62iT - 73T^{2} $
	79	$ 1 - 4.91T + 79T^{2} $
	83	$ 1 + 10.3T + 83T^{2} $
	89	$ 1 + 12.9T + 89T^{2} $
	97	$ 1 - 4.60T + 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

−8.798530902840110518486037243649, −7.86750387482738919025157092066, −7.11048716010049902390093093883, −6.33359552653525661334760088269, −5.50522099737671647616209824692, −4.94845803767096239306602006591, −4.17112864984822609948844810160, −2.57677632317893657515815389498, −2.08972092321898283649317041324, −1.09457972611112948390443081280, 1.46143660557126922811598234409, 1.72759094300985484192400350724, 2.96749201552675547429038775703, 4.08444210041754274359545067154, 5.07836673301544125817236853328, 5.68295688427955764507017696830, 6.30145011778615249913629619271, 7.06627447628439570019050884829, 8.212452091083253534545902207111, 8.775578617728810185180027253396

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(3.215947671\)
\(L(\frac12)\)	\(\approx\)	\(3.215947671\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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