↑

Properties

Label 2-1008-28.27-c1-0-11
Degree $2$
Conductor $1008$
Sign $0.944 + 0.327i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 1.73i)7-s + 3.46i·11-s − 6.92i·17-s + 4·19-s + 3.46i·23-s + 5·25-s + 6·29-s − 4·31-s − 2·37-s − 6.92i·41-s + 3.46i·43-s + (1.00 − 6.92i)49-s + 6·53-s + 12·59-s − 13.8i·61-s + ⋯
L(s)  = 1  + (0.755 − 0.654i)7-s + 1.04i·11-s − 1.68i·17-s + 0.917·19-s + 0.722i·23-s + 25-s + 1.11·29-s − 0.718·31-s − 0.328·37-s − 1.08i·41-s + 0.528i·43-s + (0.142 − 0.989i)49-s + 0.824·53-s + 1.56·59-s − 1.77i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.944 + 0.327i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.944 + 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.785195623\)
\(L(\frac12)\) \(\approx\) \(1.785195623\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-2 + 1.73i)T \)
good5 \( 1 - 5T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 6.92iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 - 3.46iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 - 13.8iT - 73T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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

−9.871981688195389682261978713353, −9.227829720851728686081192994451, −8.160257545778342274899142776482, −7.24833176421315607991797872438, −6.91212320169371939248477458813, −5.27690904928570667651465073134, −4.83880530517416181172016645013, −3.68074007176190163819888254170, −2.41171090516796137271809205657, −1.03500118034474957187957702777, 1.23343893795852360722310466749, 2.59340766035023290254891504314, 3.70055425777056835726827765200, 4.86216118013602319790804968567, 5.71732302996958553277482985995, 6.48030015107430280968112453566, 7.65624467973329127407607868297, 8.582377395619179432081028745754, 8.802081183446129283438840395423, 10.18157633524523136642510463533

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