Properties

Label 2-384-1.1-c7-0-3
Degree $2$
Conductor $384$
Sign $1$
Analytic cond. $119.955$
Root an. cond. $10.9524$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 27·3-s − 432.·5-s + 113.·7-s + 729·9-s − 4.05e3·11-s − 1.27e4·13-s − 1.16e4·15-s − 3.71e4·17-s − 4.36e4·19-s + 3.06e3·21-s + 1.06e5·23-s + 1.08e5·25-s + 1.96e4·27-s + 374.·29-s + 1.76e5·31-s − 1.09e5·33-s − 4.91e4·35-s − 2.56e5·37-s − 3.43e5·39-s − 8.53e5·41-s + 1.37e5·43-s − 3.15e5·45-s + 1.05e6·47-s − 8.10e5·49-s − 1.00e6·51-s + 1.02e6·53-s + 1.75e6·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.54·5-s + 0.125·7-s + 0.333·9-s − 0.918·11-s − 1.60·13-s − 0.893·15-s − 1.83·17-s − 1.46·19-s + 0.0722·21-s + 1.81·23-s + 1.39·25-s + 0.192·27-s + 0.00285·29-s + 1.06·31-s − 0.530·33-s − 0.193·35-s − 0.832·37-s − 0.928·39-s − 1.93·41-s + 0.264·43-s − 0.515·45-s + 1.48·47-s − 0.984·49-s − 1.05·51-s + 0.942·53-s + 1.42·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $1$
Analytic conductor: \(119.955\)
Root analytic conductor: \(10.9524\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.6176664404\)
\(L(\frac12)\) \(\approx\) \(0.6176664404\)
\(L(\frac{9}{2})\) 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 \( 1 - 27T \)
good5 \( 1 + 432.T + 7.81e4T^{2} \)
7 \( 1 - 113.T + 8.23e5T^{2} \)
11 \( 1 + 4.05e3T + 1.94e7T^{2} \)
13 \( 1 + 1.27e4T + 6.27e7T^{2} \)
17 \( 1 + 3.71e4T + 4.10e8T^{2} \)
19 \( 1 + 4.36e4T + 8.93e8T^{2} \)
23 \( 1 - 1.06e5T + 3.40e9T^{2} \)
29 \( 1 - 374.T + 1.72e10T^{2} \)
31 \( 1 - 1.76e5T + 2.75e10T^{2} \)
37 \( 1 + 2.56e5T + 9.49e10T^{2} \)
41 \( 1 + 8.53e5T + 1.94e11T^{2} \)
43 \( 1 - 1.37e5T + 2.71e11T^{2} \)
47 \( 1 - 1.05e6T + 5.06e11T^{2} \)
53 \( 1 - 1.02e6T + 1.17e12T^{2} \)
59 \( 1 + 6.39e5T + 2.48e12T^{2} \)
61 \( 1 + 1.60e6T + 3.14e12T^{2} \)
67 \( 1 - 1.63e6T + 6.06e12T^{2} \)
71 \( 1 + 1.31e6T + 9.09e12T^{2} \)
73 \( 1 + 2.28e6T + 1.10e13T^{2} \)
79 \( 1 + 7.00e6T + 1.92e13T^{2} \)
83 \( 1 - 5.53e6T + 2.71e13T^{2} \)
89 \( 1 - 2.52e6T + 4.42e13T^{2} \)
97 \( 1 - 4.71e6T + 8.07e13T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.30186562070331896921199161893, −8.903800060395534217966560157410, −8.373075063287095037987886090878, −7.37114360713870961262295863279, −6.81602393966748388881256814089, −4.85975831524861752069157083430, −4.40156698091464796058795630452, −3.07335338671218980505697670137, −2.20588135183470170732372068306, −0.33096069031303393593107961324, 0.33096069031303393593107961324, 2.20588135183470170732372068306, 3.07335338671218980505697670137, 4.40156698091464796058795630452, 4.85975831524861752069157083430, 6.81602393966748388881256814089, 7.37114360713870961262295863279, 8.373075063287095037987886090878, 8.903800060395534217966560157410, 10.30186562070331896921199161893

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