↑

Properties

Label 2-2700-5.4-c1-0-18
Degree $2$
Conductor $2700$
Sign $-0.894 + 0.447i$
Analytic cond. $21.5596$
Root an. cond. $4.64323$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4i·7-s − 6·11-s − 4i·13-s − 3i·17-s + 7·19-s + 9i·23-s − 7·31-s − 2i·37-s − 6·41-s + 2i·43-s − 9·49-s − 9i·53-s − 12·59-s − 7·61-s − 2i·67-s + ⋯
L(s)  = 1  + 1.51i·7-s − 1.80·11-s − 1.10i·13-s − 0.727i·17-s + 1.60·19-s + 1.87i·23-s − 1.25·31-s − 0.328i·37-s − 0.937·41-s + 0.304i·43-s − 1.28·49-s − 1.23i·53-s − 1.56·59-s − 0.896·61-s − 0.244i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(21.5596\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2700,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 - 9iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 9iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - T + 79T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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

−8.448256394453675915105094268733, −7.71787410308609866490536306678, −7.29410198358181911079706203638, −5.78790221478801988402491448720, −5.44760221716614385676235217505, −5.00431193121033736073709868254, −3.21518374093782558734163368682, −2.93551248908926758209320965442, −1.75874636376727431618080420984, 0, 1.35894413706850220747878729608, 2.60650607814006130489689815032, 3.59667474284058358323643278848, 4.47546392109458353031834745874, 5.12619567431031463841535205410, 6.17054588845563049310059357245, 7.07326090644926444111319389352, 7.56523375781758488325644225583, 8.250291085340585587247314030202

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