Properties

Label 2-384-1.1-c7-0-29
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 + 170.·5-s + 803.·7-s + 729·9-s + 8.55e3·11-s − 1.46e3·13-s − 4.59e3·15-s + 3.07e4·17-s + 1.13e4·19-s − 2.16e4·21-s + 1.16e5·23-s − 4.91e4·25-s − 1.96e4·27-s + 1.84e5·29-s − 6.13e4·31-s − 2.31e5·33-s + 1.36e5·35-s + 3.57e5·37-s + 3.95e4·39-s + 3.89e5·41-s − 9.39e5·43-s + 1.24e5·45-s − 1.13e6·47-s − 1.78e5·49-s − 8.31e5·51-s + 3.24e5·53-s + 1.45e6·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.609·5-s + 0.885·7-s + 0.333·9-s + 1.93·11-s − 0.185·13-s − 0.351·15-s + 1.51·17-s + 0.381·19-s − 0.511·21-s + 1.99·23-s − 0.628·25-s − 0.192·27-s + 1.40·29-s − 0.369·31-s − 1.11·33-s + 0.539·35-s + 1.15·37-s + 0.106·39-s + 0.882·41-s − 1.80·43-s + 0.203·45-s − 1.60·47-s − 0.216·49-s − 0.877·51-s + 0.299·53-s + 1.18·55-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

\(L(4)\) \(\approx\) \(3.267436348\)
\(L(\frac12)\) \(\approx\) \(3.267436348\)
\(L(\frac{9}{2})\) 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 \( 1 - 170.T + 7.81e4T^{2} \)
7 \( 1 - 803.T + 8.23e5T^{2} \)
11 \( 1 - 8.55e3T + 1.94e7T^{2} \)
13 \( 1 + 1.46e3T + 6.27e7T^{2} \)
17 \( 1 - 3.07e4T + 4.10e8T^{2} \)
19 \( 1 - 1.13e4T + 8.93e8T^{2} \)
23 \( 1 - 1.16e5T + 3.40e9T^{2} \)
29 \( 1 - 1.84e5T + 1.72e10T^{2} \)
31 \( 1 + 6.13e4T + 2.75e10T^{2} \)
37 \( 1 - 3.57e5T + 9.49e10T^{2} \)
41 \( 1 - 3.89e5T + 1.94e11T^{2} \)
43 \( 1 + 9.39e5T + 2.71e11T^{2} \)
47 \( 1 + 1.13e6T + 5.06e11T^{2} \)
53 \( 1 - 3.24e5T + 1.17e12T^{2} \)
59 \( 1 - 1.86e6T + 2.48e12T^{2} \)
61 \( 1 + 1.14e6T + 3.14e12T^{2} \)
67 \( 1 + 1.58e6T + 6.06e12T^{2} \)
71 \( 1 + 3.98e6T + 9.09e12T^{2} \)
73 \( 1 - 3.61e6T + 1.10e13T^{2} \)
79 \( 1 + 4.26e6T + 1.92e13T^{2} \)
83 \( 1 - 1.61e5T + 2.71e13T^{2} \)
89 \( 1 + 9.25e6T + 4.42e13T^{2} \)
97 \( 1 + 8.61e6T + 8.07e13T^{2} \)
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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

−10.04873690752787356634212108481, −9.413647770713992271749469424883, −8.350372669870020655361389219923, −7.19004249570019973078761284713, −6.32585497882127143005424700207, −5.35495398909174766470442178143, −4.46952553616210915080110357620, −3.18289653470736433395013161993, −1.51520431439735103897057654735, −1.02358244873741824846200723606, 1.02358244873741824846200723606, 1.51520431439735103897057654735, 3.18289653470736433395013161993, 4.46952553616210915080110357620, 5.35495398909174766470442178143, 6.32585497882127143005424700207, 7.19004249570019973078761284713, 8.350372669870020655361389219923, 9.413647770713992271749469424883, 10.04873690752787356634212108481

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