↑

Properties

Label 2-4800-5.4-c1-0-49
Degree $2$
Conductor $4800$
Sign $-0.447 + 0.894i$
Analytic cond. $38.3281$
Root an. cond. $6.19097$
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  + i·3-s + i·7-s − 9-s − 4·11-s + 3i·13-s + 4i·17-s + 19-s − 21-s − i·27-s − 8·29-s − 31-s − 4i·33-s + 2i·37-s − 3·39-s + 2·41-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.377i·7-s − 0.333·9-s − 1.20·11-s + 0.832i·13-s + 0.970i·17-s + 0.229·19-s − 0.218·21-s − 0.192i·27-s − 1.48·29-s − 0.179·31-s − 0.696i·33-s + 0.328i·37-s − 0.480·39-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(38.3281\)
Root analytic conductor: \(6.19097\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4800} (3649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 4800,\ (\ :1/2),\ -0.447 + 0.894i)\)

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 \( 1 - iT \)
5 \( 1 \)
good7 \( 1 - iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 3iT - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 11iT - 43T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 11T + 61T^{2} \)
67 \( 1 + 9iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 14iT - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 - 2iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 11iT - 97T^{2} \)
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

−8.006280407388450515699716156338, −7.49063598332903952567133249923, −6.51302994311780862020128124333, −5.68870526685814841129496101740, −5.20224596787694928165605042541, −4.26199063874757275686652928322, −3.57831473662936926002397060241, −2.57626078884835906493365862569, −1.74926394739310545593824860264, 0, 1.07296677050852244362024250789, 2.34490635738277151214968333739, 2.98852411186726859630498117633, 3.95355910551855508780510555330, 5.08524307547232825946154577914, 5.49516321675733517774106995499, 6.37741633054757258228994031045, 7.35730543973915016802630238906, 7.61382570906626423871747880025

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