Properties

Label 2-384-1.1-c1-0-2
Degree $2$
Conductor $384$
Sign $1$
Analytic cond. $3.06625$
Root an. cond. $1.75107$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4·5-s − 2·7-s + 9-s + 4·11-s − 2·13-s − 4·15-s − 2·17-s + 8·19-s + 2·21-s − 4·23-s + 11·25-s − 27-s + 6·31-s − 4·33-s − 8·35-s + 2·37-s + 2·39-s + 6·41-s + 4·45-s − 4·47-s − 3·49-s + 2·51-s + 16·55-s − 8·57-s − 4·59-s − 14·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 1.03·15-s − 0.485·17-s + 1.83·19-s + 0.436·21-s − 0.834·23-s + 11/5·25-s − 0.192·27-s + 1.07·31-s − 0.696·33-s − 1.35·35-s + 0.328·37-s + 0.320·39-s + 0.937·41-s + 0.596·45-s − 0.583·47-s − 3/7·49-s + 0.280·51-s + 2.15·55-s − 1.05·57-s − 0.520·59-s − 1.79·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $1$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{384} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.483899732\)
\(L(\frac12)\) \(\approx\) \(1.483899732\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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

−11.38766792869140056495258951368, −10.06632064709018516311308621993, −9.738751180865856613138829349946, −8.985911500488081519481984584802, −7.27371486188202795087403560747, −6.30170361535159036673173480205, −5.81778647559970487461786585347, −4.60632905768716122167488407077, −2.92708427914109016867182117304, −1.43950309180346771366044691508, 1.43950309180346771366044691508, 2.92708427914109016867182117304, 4.60632905768716122167488407077, 5.81778647559970487461786585347, 6.30170361535159036673173480205, 7.27371486188202795087403560747, 8.985911500488081519481984584802, 9.738751180865856613138829349946, 10.06632064709018516311308621993, 11.38766792869140056495258951368

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