Properties

Label 2-384-1.1-c7-0-13
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 − 283.·5-s + 473.·7-s + 729·9-s − 1.65e3·11-s + 507.·13-s − 7.65e3·15-s − 1.34e4·17-s + 9.26e3·19-s + 1.27e4·21-s + 5.49e3·23-s + 2.26e3·25-s + 1.96e4·27-s + 1.72e5·29-s − 1.62e5·31-s − 4.47e4·33-s − 1.34e5·35-s + 2.39e5·37-s + 1.37e4·39-s + 1.79e5·41-s − 6.92e5·43-s − 2.06e5·45-s − 7.30e4·47-s − 5.99e5·49-s − 3.63e5·51-s + 9.70e5·53-s + 4.69e5·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.01·5-s + 0.522·7-s + 0.333·9-s − 0.375·11-s + 0.0640·13-s − 0.585·15-s − 0.664·17-s + 0.310·19-s + 0.301·21-s + 0.0942·23-s + 0.0289·25-s + 0.192·27-s + 1.31·29-s − 0.979·31-s − 0.216·33-s − 0.529·35-s + 0.776·37-s + 0.0369·39-s + 0.407·41-s − 1.32·43-s − 0.338·45-s − 0.102·47-s − 0.727·49-s − 0.383·51-s + 0.895·53-s + 0.380·55-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

\(L(4)\) \(\approx\) \(2.069453208\)
\(L(\frac12)\) \(\approx\) \(2.069453208\)
\(L(\frac{9}{2})\) 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 \( 1 + 283.T + 7.81e4T^{2} \)
7 \( 1 - 473.T + 8.23e5T^{2} \)
11 \( 1 + 1.65e3T + 1.94e7T^{2} \)
13 \( 1 - 507.T + 6.27e7T^{2} \)
17 \( 1 + 1.34e4T + 4.10e8T^{2} \)
19 \( 1 - 9.26e3T + 8.93e8T^{2} \)
23 \( 1 - 5.49e3T + 3.40e9T^{2} \)
29 \( 1 - 1.72e5T + 1.72e10T^{2} \)
31 \( 1 + 1.62e5T + 2.75e10T^{2} \)
37 \( 1 - 2.39e5T + 9.49e10T^{2} \)
41 \( 1 - 1.79e5T + 1.94e11T^{2} \)
43 \( 1 + 6.92e5T + 2.71e11T^{2} \)
47 \( 1 + 7.30e4T + 5.06e11T^{2} \)
53 \( 1 - 9.70e5T + 1.17e12T^{2} \)
59 \( 1 - 4.16e5T + 2.48e12T^{2} \)
61 \( 1 - 1.25e5T + 3.14e12T^{2} \)
67 \( 1 + 6.64e5T + 6.06e12T^{2} \)
71 \( 1 - 2.50e6T + 9.09e12T^{2} \)
73 \( 1 - 1.65e6T + 1.10e13T^{2} \)
79 \( 1 - 3.00e6T + 1.92e13T^{2} \)
83 \( 1 - 4.05e6T + 2.71e13T^{2} \)
89 \( 1 - 7.36e6T + 4.42e13T^{2} \)
97 \( 1 + 7.93e6T + 8.07e13T^{2} \)
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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

−10.17376121571962756474350553841, −9.054399549502015862062298861073, −8.194617841807431745306104222557, −7.61621109905871055236890567292, −6.57559756402578429756935215742, −5.10116121464551122188651401782, −4.19422022753022250126235347002, −3.20789761637468074919809984634, −2.02080670209711173544830168389, −0.64254868682927879166074100725, 0.64254868682927879166074100725, 2.02080670209711173544830168389, 3.20789761637468074919809984634, 4.19422022753022250126235347002, 5.10116121464551122188651401782, 6.57559756402578429756935215742, 7.61621109905871055236890567292, 8.194617841807431745306104222557, 9.054399549502015862062298861073, 10.17376121571962756474350553841

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