Properties

Label 2-384-1.1-c9-0-48
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 + 1.65e3·5-s − 1.13e4·7-s + 6.56e3·9-s + 4.38e4·11-s − 2.01e4·13-s − 1.34e5·15-s + 1.07e5·17-s − 8.70e5·19-s + 9.20e5·21-s + 1.61e6·23-s + 7.95e5·25-s − 5.31e5·27-s − 7.11e5·29-s + 4.84e6·31-s − 3.54e6·33-s − 1.88e7·35-s + 3.54e5·37-s + 1.63e6·39-s + 2.36e7·41-s − 5.28e6·43-s + 1.08e7·45-s − 5.51e7·47-s + 8.89e7·49-s − 8.73e6·51-s + 5.31e7·53-s + 7.26e7·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.18·5-s − 1.78·7-s + 0.333·9-s + 0.902·11-s − 0.196·13-s − 0.684·15-s + 0.313·17-s − 1.53·19-s + 1.03·21-s + 1.20·23-s + 0.407·25-s − 0.192·27-s − 0.186·29-s + 0.941·31-s − 0.521·33-s − 2.12·35-s + 0.0310·37-s + 0.113·39-s + 1.30·41-s − 0.235·43-s + 0.395·45-s − 1.64·47-s + 2.20·49-s − 0.180·51-s + 0.925·53-s + 1.07·55-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 \( 1 - 1.65e3T + 1.95e6T^{2} \)
7 \( 1 + 1.13e4T + 4.03e7T^{2} \)
11 \( 1 - 4.38e4T + 2.35e9T^{2} \)
13 \( 1 + 2.01e4T + 1.06e10T^{2} \)
17 \( 1 - 1.07e5T + 1.18e11T^{2} \)
19 \( 1 + 8.70e5T + 3.22e11T^{2} \)
23 \( 1 - 1.61e6T + 1.80e12T^{2} \)
29 \( 1 + 7.11e5T + 1.45e13T^{2} \)
31 \( 1 - 4.84e6T + 2.64e13T^{2} \)
37 \( 1 - 3.54e5T + 1.29e14T^{2} \)
41 \( 1 - 2.36e7T + 3.27e14T^{2} \)
43 \( 1 + 5.28e6T + 5.02e14T^{2} \)
47 \( 1 + 5.51e7T + 1.11e15T^{2} \)
53 \( 1 - 5.31e7T + 3.29e15T^{2} \)
59 \( 1 + 1.35e8T + 8.66e15T^{2} \)
61 \( 1 + 2.20e7T + 1.16e16T^{2} \)
67 \( 1 - 1.20e8T + 2.72e16T^{2} \)
71 \( 1 + 1.30e8T + 4.58e16T^{2} \)
73 \( 1 - 2.81e7T + 5.88e16T^{2} \)
79 \( 1 + 5.16e8T + 1.19e17T^{2} \)
83 \( 1 - 5.45e8T + 1.86e17T^{2} \)
89 \( 1 + 1.34e8T + 3.50e17T^{2} \)
97 \( 1 + 1.29e9T + 7.60e17T^{2} \)
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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

−9.572772664324513886974114594486, −8.777532601354599077505371808198, −7.07927852422876537833622015792, −6.31198422905612158357922018632, −5.95203174342194529914137030036, −4.60364643630617710109456710164, −3.39185306807798248091796413595, −2.32992353924345293166665294498, −1.10264839319436809075677086964, 0, 1.10264839319436809075677086964, 2.32992353924345293166665294498, 3.39185306807798248091796413595, 4.60364643630617710109456710164, 5.95203174342194529914137030036, 6.31198422905612158357922018632, 7.07927852422876537833622015792, 8.777532601354599077505371808198, 9.572772664324513886974114594486

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