Properties

Label 2-5400-5.4-c1-0-44
Degree $2$
Conductor $5400$
Sign $0.447 + 0.894i$
Analytic cond. $43.1192$
Root an. cond. $6.56652$
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  + 3i·7-s − 4·11-s + i·13-s − 4i·17-s + 19-s − 4i·23-s − 4·31-s + 9i·37-s − 8i·43-s − 12i·47-s − 2·49-s + 8i·53-s + 4·59-s − 5·61-s − 11i·67-s + ⋯
L(s)  = 1  + 1.13i·7-s − 1.20·11-s + 0.277i·13-s − 0.970i·17-s + 0.229·19-s − 0.834i·23-s − 0.718·31-s + 1.47i·37-s − 1.21i·43-s − 1.75i·47-s − 0.285·49-s + 1.09i·53-s + 0.520·59-s − 0.640·61-s − 1.34i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5400\)    =    \(2^{3} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(43.1192\)
Root analytic conductor: \(6.56652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5400} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5400,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.128459145\)
\(L(\frac12)\) \(\approx\) \(1.128459145\)
\(L(\frac{3}{2})\) 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 \( 1 - 3iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 9iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 - 8iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 5T + 61T^{2} \)
67 \( 1 + 11iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - iT - 73T^{2} \)
79 \( 1 - 5T + 79T^{2} \)
83 \( 1 + 8iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + 5iT - 97T^{2} \)
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

−8.086592132677921025191161548005, −7.37024241375507521877431021002, −6.63199738985647254933871627640, −5.75271000851018500666721532981, −5.19506992395321585401884981296, −4.57715545261719796054605469987, −3.34208327373457238836778933890, −2.63210187062134322044630878622, −1.92280245962109758082073944973, −0.34392919915695631528291515964, 0.921132347323204340622513395384, 2.01611707602557258234476057677, 3.09438724081690456112133826257, 3.83120707306438466663407756064, 4.59033382765709404948124791056, 5.46253275237421675685337453950, 6.06042880226455265433924219883, 7.06303904397808918582503602544, 7.64908150247277608003320668676, 8.041764823393839272873227907413

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