↑

Properties

Label 2-3528-504.227-c0-0-5
Degree $2$
Conductor $3528$
Sign $0.0810 - 0.996i$
Analytic cond. $1.76070$
Root an. cond. $1.32691$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + (0.793 + 0.608i)3-s − 4-s + (−0.608 + 0.793i)6-s − i·8-s + (0.258 + 0.965i)9-s + (1.67 − 0.965i)11-s + (−0.793 − 0.608i)12-s + 16-s + (0.608 − 1.05i)17-s + (−0.965 + 0.258i)18-s + (0.226 − 0.130i)19-s + (0.965 + 1.67i)22-s + (0.608 − 0.793i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + i·2-s + (0.793 + 0.608i)3-s − 4-s + (−0.608 + 0.793i)6-s − i·8-s + (0.258 + 0.965i)9-s + (1.67 − 0.965i)11-s + (−0.793 − 0.608i)12-s + 16-s + (0.608 − 1.05i)17-s + (−0.965 + 0.258i)18-s + (0.226 − 0.130i)19-s + (0.965 + 1.67i)22-s + (0.608 − 0.793i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0810 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0810 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.0810 - 0.996i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.0810 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.703819974\)
\(L(\frac12)\) \(\approx\) \(1.703819974\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.793 - 0.608i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.608 + 1.05i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.226 + 0.130i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.793 - 1.37i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + 1.98T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 1.73T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.71 + 0.991i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.37 + 0.793i)T + (0.5 + 0.866i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.953722995683530475367054510053, −8.153819996581661022872225281364, −7.60378373225246490479789261465, −6.69097342572239612570075605209, −6.02814694274480882141305449417, −5.13689181285931378121520585788, −4.30687430356338712848028624443, −3.67219783360426379146716796201, −2.83867361135899845781541256415, −1.18044967447081141471602298490, 1.34193865002238550815484174361, 1.80631224634585544729146335804, 2.95119436357343806395754169136, 3.87171417901966812366909570364, 4.21086465340203987693816219778, 5.54022927510132382059752452661, 6.39071826287132221410161925339, 7.29562499902007176678505313711, 7.935570413479033118239945647626, 8.798522340391182089187461132771

Graph of the $Z$-function along the critical line