Properties

Label 2-3840-120.29-c0-0-4
Degree $2$
Conductor $3840$
Sign $1$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
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  + (0.707 + 0.707i)3-s − 5-s − 1.41i·7-s + 1.00i·9-s + (−0.707 − 0.707i)15-s + (1.00 − 1.00i)21-s + 1.41·23-s + 25-s + (−0.707 + 0.707i)27-s + 1.41i·35-s − 2i·41-s + 1.41·43-s − 1.00i·45-s + 1.41·47-s − 1.00·49-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s − 5-s − 1.41i·7-s + 1.00i·9-s + (−0.707 − 0.707i)15-s + (1.00 − 1.00i)21-s + 1.41·23-s + 25-s + (−0.707 + 0.707i)27-s + 1.41i·35-s − 2i·41-s + 1.41·43-s − 1.00i·45-s + 1.41·47-s − 1.00·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $1$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.391697130\)
\(L(\frac12)\) \(\approx\) \(1.391697130\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + T \)
good7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + 2iT - T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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

−8.774841428552050549519257302023, −7.80088564320645050970796679859, −7.40519660058103081992144492389, −6.80014998803149132875143788165, −5.42832202003097371252154260544, −4.61360348343072813955555176988, −3.96092236669099698664225630391, −3.48764252977294784069339095837, −2.47093531157134890589620554362, −0.907516935546230755221536131392, 1.12418347067374407920736451455, 2.46337935622586281392649732504, 2.96095573817742705856300078850, 3.90952483144489179310417362983, 4.88684051379152758422520559656, 5.78808989137020646873215763814, 6.60417902498538967695676089779, 7.31689532647479686569068978226, 7.996603816067804511440668886636, 8.639163022862277360086964484793

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