Properties

Label 2-384-1.1-c7-0-0
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 − 127.·5-s − 547.·7-s + 729·9-s − 5.22e3·11-s − 1.10e4·13-s + 3.43e3·15-s + 3.56e3·17-s − 3.78e4·19-s + 1.47e4·21-s + 3.18e3·23-s − 6.19e4·25-s − 1.96e4·27-s + 3.65e3·29-s − 1.81e5·31-s + 1.41e5·33-s + 6.96e4·35-s + 1.33e5·37-s + 2.97e5·39-s − 1.89e5·41-s − 4.33e5·43-s − 9.28e4·45-s − 1.27e5·47-s − 5.24e5·49-s − 9.61e4·51-s − 1.84e6·53-s + 6.65e5·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.455·5-s − 0.603·7-s + 0.333·9-s − 1.18·11-s − 1.39·13-s + 0.262·15-s + 0.175·17-s − 1.26·19-s + 0.348·21-s + 0.0546·23-s − 0.792·25-s − 0.192·27-s + 0.0278·29-s − 1.09·31-s + 0.683·33-s + 0.274·35-s + 0.434·37-s + 0.802·39-s − 0.430·41-s − 0.831·43-s − 0.151·45-s − 0.179·47-s − 0.636·49-s − 0.101·51-s − 1.70·53-s + 0.539·55-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

\(L(4)\) \(\approx\) \(0.03357397390\)
\(L(\frac12)\) \(\approx\) \(0.03357397390\)
\(L(\frac{9}{2})\) 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 \( 1 + 127.T + 7.81e4T^{2} \)
7 \( 1 + 547.T + 8.23e5T^{2} \)
11 \( 1 + 5.22e3T + 1.94e7T^{2} \)
13 \( 1 + 1.10e4T + 6.27e7T^{2} \)
17 \( 1 - 3.56e3T + 4.10e8T^{2} \)
19 \( 1 + 3.78e4T + 8.93e8T^{2} \)
23 \( 1 - 3.18e3T + 3.40e9T^{2} \)
29 \( 1 - 3.65e3T + 1.72e10T^{2} \)
31 \( 1 + 1.81e5T + 2.75e10T^{2} \)
37 \( 1 - 1.33e5T + 9.49e10T^{2} \)
41 \( 1 + 1.89e5T + 1.94e11T^{2} \)
43 \( 1 + 4.33e5T + 2.71e11T^{2} \)
47 \( 1 + 1.27e5T + 5.06e11T^{2} \)
53 \( 1 + 1.84e6T + 1.17e12T^{2} \)
59 \( 1 - 4.49e5T + 2.48e12T^{2} \)
61 \( 1 + 3.57e5T + 3.14e12T^{2} \)
67 \( 1 + 4.13e6T + 6.06e12T^{2} \)
71 \( 1 + 4.53e6T + 9.09e12T^{2} \)
73 \( 1 - 5.45e6T + 1.10e13T^{2} \)
79 \( 1 - 2.20e6T + 1.92e13T^{2} \)
83 \( 1 + 5.08e5T + 2.71e13T^{2} \)
89 \( 1 + 9.79e5T + 4.42e13T^{2} \)
97 \( 1 + 4.99e6T + 8.07e13T^{2} \)
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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

−10.21855737068131802407715993657, −9.443952608861145170298424155060, −8.112294367345583308441449975182, −7.36870430490234901285583273472, −6.36528240750520531786090191091, −5.30204884396201608139563361442, −4.42407594599404673328467001412, −3.13067269673774481599959378820, −1.97171917795230638677856483641, −0.081814977589238039949477988126, 0.081814977589238039949477988126, 1.97171917795230638677856483641, 3.13067269673774481599959378820, 4.42407594599404673328467001412, 5.30204884396201608139563361442, 6.36528240750520531786090191091, 7.36870430490234901285583273472, 8.112294367345583308441449975182, 9.443952608861145170298424155060, 10.21855737068131802407715993657

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