↑

Properties

Label 2-3528-504.331-c0-0-4
Degree $2$
Conductor $3528$
Sign $0.678 + 0.734i$
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  + (−0.5 − 0.866i)2-s + 3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)6-s + 0.999·8-s + 9-s − 11-s + (−0.499 + 0.866i)12-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)22-s + 0.999·24-s + 25-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)6-s + 0.999·8-s + 9-s − 11-s + (−0.499 + 0.866i)12-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)22-s + 0.999·24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.678 + 0.734i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (1843, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.678 + 0.734i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.379549113\)
\(L(\frac12)\) \(\approx\) \(1.379549113\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 - T \)
7 \( 1 \)
good5 \( 1 - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.547535075482230372448252642613, −8.233119293747734303473732040272, −7.45185699674951161027977607712, −6.79783839909759789691453592979, −5.39258635330910742295544068833, −4.58160140798165828307024995080, −3.69586198249618634634450616428, −2.91766430332428793833333450952, −2.27733803070679252275354826826, −1.12163739915142526358635137154, 1.12439516224268404002015935887, 2.36735391477747410644149746966, 3.29373435423662455902765851166, 4.38096589150265543160034273227, 5.15146183548959411781646441420, 5.90174901299910401048398108363, 6.95460098476231745326507669379, 7.53672188398300010710615397531, 7.987470978298087156665607743982, 8.791039308809892399410226926121

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