Properties

Label 2-384-1.1-c9-0-71
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 81·3-s + 2.68e3·5-s + 950.·7-s + 6.56e3·9-s − 8.53e4·11-s − 1.24e5·13-s + 2.17e5·15-s − 2.14e4·17-s + 2.43e5·19-s + 7.69e4·21-s − 7.06e5·23-s + 5.26e6·25-s + 5.31e5·27-s − 3.90e6·29-s + 8.04e6·31-s − 6.91e6·33-s + 2.55e6·35-s − 1.67e7·37-s − 1.01e7·39-s − 3.31e7·41-s + 3.30e7·43-s + 1.76e7·45-s − 3.29e7·47-s − 3.94e7·49-s − 1.73e6·51-s − 5.61e7·53-s − 2.29e8·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.92·5-s + 0.149·7-s + 0.333·9-s − 1.75·11-s − 1.21·13-s + 1.11·15-s − 0.0622·17-s + 0.427·19-s + 0.0863·21-s − 0.526·23-s + 2.69·25-s + 0.192·27-s − 1.02·29-s + 1.56·31-s − 1.01·33-s + 0.287·35-s − 1.46·37-s − 0.700·39-s − 1.83·41-s + 1.47·43-s + 0.640·45-s − 0.985·47-s − 0.977·49-s − 0.0359·51-s − 0.978·53-s − 3.38·55-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: \(1\)
Selberg data: \((2,\ 384,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 - 2.68e3T + 1.95e6T^{2} \)
7 \( 1 - 950.T + 4.03e7T^{2} \)
11 \( 1 + 8.53e4T + 2.35e9T^{2} \)
13 \( 1 + 1.24e5T + 1.06e10T^{2} \)
17 \( 1 + 2.14e4T + 1.18e11T^{2} \)
19 \( 1 - 2.43e5T + 3.22e11T^{2} \)
23 \( 1 + 7.06e5T + 1.80e12T^{2} \)
29 \( 1 + 3.90e6T + 1.45e13T^{2} \)
31 \( 1 - 8.04e6T + 2.64e13T^{2} \)
37 \( 1 + 1.67e7T + 1.29e14T^{2} \)
41 \( 1 + 3.31e7T + 3.27e14T^{2} \)
43 \( 1 - 3.30e7T + 5.02e14T^{2} \)
47 \( 1 + 3.29e7T + 1.11e15T^{2} \)
53 \( 1 + 5.61e7T + 3.29e15T^{2} \)
59 \( 1 - 1.09e7T + 8.66e15T^{2} \)
61 \( 1 + 3.77e7T + 1.16e16T^{2} \)
67 \( 1 + 9.13e7T + 2.72e16T^{2} \)
71 \( 1 - 2.44e8T + 4.58e16T^{2} \)
73 \( 1 - 1.30e8T + 5.88e16T^{2} \)
79 \( 1 + 4.26e8T + 1.19e17T^{2} \)
83 \( 1 - 2.32e8T + 1.86e17T^{2} \)
89 \( 1 + 2.89e8T + 3.50e17T^{2} \)
97 \( 1 + 2.74e8T + 7.60e17T^{2} \)
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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

−9.659872135406217489885895933184, −8.497778906215394494582340144224, −7.56733435492478340802102887343, −6.50768238979770726416168102148, −5.36807796612241096167530796992, −4.90570361634913085024799041745, −3.01560865572331279762419658956, −2.33906356220073651852521195366, −1.58012040536872639645512883543, 0, 1.58012040536872639645512883543, 2.33906356220073651852521195366, 3.01560865572331279762419658956, 4.90570361634913085024799041745, 5.36807796612241096167530796992, 6.50768238979770726416168102148, 7.56733435492478340802102887343, 8.497778906215394494582340144224, 9.659872135406217489885895933184

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