Properties

Label 2-384-1.1-c9-0-37
Degree $2$
Conductor $384$
Sign $1$
Analytic cond. $197.773$
Root an. cond. $14.0632$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 81·3-s + 2.12e3·5-s − 1.44e3·7-s + 6.56e3·9-s − 2.01e4·11-s + 1.50e5·13-s + 1.72e5·15-s + 3.51e5·17-s + 5.84e5·19-s − 1.17e5·21-s − 2.89e5·23-s + 2.56e6·25-s + 5.31e5·27-s + 2.55e6·29-s − 9.90e6·31-s − 1.63e6·33-s − 3.07e6·35-s + 3.78e6·37-s + 1.21e7·39-s + 1.02e7·41-s + 3.60e6·43-s + 1.39e7·45-s − 1.06e7·47-s − 3.82e7·49-s + 2.84e7·51-s + 4.14e7·53-s − 4.28e7·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.52·5-s − 0.227·7-s + 0.333·9-s − 0.415·11-s + 1.45·13-s + 0.878·15-s + 1.02·17-s + 1.02·19-s − 0.131·21-s − 0.215·23-s + 1.31·25-s + 0.192·27-s + 0.670·29-s − 1.92·31-s − 0.239·33-s − 0.346·35-s + 0.332·37-s + 0.841·39-s + 0.567·41-s + 0.160·43-s + 0.507·45-s − 0.317·47-s − 0.948·49-s + 0.589·51-s + 0.720·53-s − 0.631·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $1$
Analytic conductor: \(197.773\)
Root analytic conductor: \(14.0632\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(4.868730100\)
\(L(\frac12)\) \(\approx\) \(4.868730100\)
\(L(\frac{11}{2})\) 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 \( 1 - 81T \)
good5 \( 1 - 2.12e3T + 1.95e6T^{2} \)
7 \( 1 + 1.44e3T + 4.03e7T^{2} \)
11 \( 1 + 2.01e4T + 2.35e9T^{2} \)
13 \( 1 - 1.50e5T + 1.06e10T^{2} \)
17 \( 1 - 3.51e5T + 1.18e11T^{2} \)
19 \( 1 - 5.84e5T + 3.22e11T^{2} \)
23 \( 1 + 2.89e5T + 1.80e12T^{2} \)
29 \( 1 - 2.55e6T + 1.45e13T^{2} \)
31 \( 1 + 9.90e6T + 2.64e13T^{2} \)
37 \( 1 - 3.78e6T + 1.29e14T^{2} \)
41 \( 1 - 1.02e7T + 3.27e14T^{2} \)
43 \( 1 - 3.60e6T + 5.02e14T^{2} \)
47 \( 1 + 1.06e7T + 1.11e15T^{2} \)
53 \( 1 - 4.14e7T + 3.29e15T^{2} \)
59 \( 1 - 9.14e7T + 8.66e15T^{2} \)
61 \( 1 - 1.51e8T + 1.16e16T^{2} \)
67 \( 1 - 1.97e8T + 2.72e16T^{2} \)
71 \( 1 + 1.32e8T + 4.58e16T^{2} \)
73 \( 1 + 1.50e8T + 5.88e16T^{2} \)
79 \( 1 + 3.92e8T + 1.19e17T^{2} \)
83 \( 1 - 4.76e7T + 1.86e17T^{2} \)
89 \( 1 + 1.00e9T + 3.50e17T^{2} \)
97 \( 1 - 8.13e8T + 7.60e17T^{2} \)
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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

−9.770814954886229754150259974668, −9.044271865302474612405188699724, −8.106233335800635623123691150675, −6.98563921649051599107551621300, −5.89723657337603336464107126391, −5.34520799575734502935854297990, −3.76139392186720169149636269383, −2.84929695064730379305111040985, −1.77485914200883655118880939514, −0.964551966257621699110950321409, 0.964551966257621699110950321409, 1.77485914200883655118880939514, 2.84929695064730379305111040985, 3.76139392186720169149636269383, 5.34520799575734502935854297990, 5.89723657337603336464107126391, 6.98563921649051599107551621300, 8.106233335800635623123691150675, 9.044271865302474612405188699724, 9.770814954886229754150259974668

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