Properties

Label 2-384-1.1-c7-0-40
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 27·3-s + 333.·5-s − 988.·7-s + 729·9-s + 4.47e3·11-s − 5.59e3·13-s − 8.99e3·15-s + 190.·17-s + 1.23e4·19-s + 2.66e4·21-s − 4.03e4·23-s + 3.29e4·25-s − 1.96e4·27-s + 6.56e4·29-s − 2.26e5·31-s − 1.20e5·33-s − 3.29e5·35-s + 6.08e5·37-s + 1.51e5·39-s + 7.87e4·41-s − 1.05e5·43-s + 2.42e5·45-s − 3.88e5·47-s + 1.53e5·49-s − 5.13e3·51-s − 5.39e4·53-s + 1.49e6·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.19·5-s − 1.08·7-s + 0.333·9-s + 1.01·11-s − 0.706·13-s − 0.688·15-s + 0.00939·17-s + 0.411·19-s + 0.628·21-s − 0.691·23-s + 0.422·25-s − 0.192·27-s + 0.499·29-s − 1.36·31-s − 0.585·33-s − 1.29·35-s + 1.97·37-s + 0.408·39-s + 0.178·41-s − 0.202·43-s + 0.397·45-s − 0.545·47-s + 0.186·49-s − 0.00542·51-s − 0.0497·53-s + 1.20·55-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: \(1\)
Selberg data: \((2,\ 384,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 - 333.T + 7.81e4T^{2} \)
7 \( 1 + 988.T + 8.23e5T^{2} \)
11 \( 1 - 4.47e3T + 1.94e7T^{2} \)
13 \( 1 + 5.59e3T + 6.27e7T^{2} \)
17 \( 1 - 190.T + 4.10e8T^{2} \)
19 \( 1 - 1.23e4T + 8.93e8T^{2} \)
23 \( 1 + 4.03e4T + 3.40e9T^{2} \)
29 \( 1 - 6.56e4T + 1.72e10T^{2} \)
31 \( 1 + 2.26e5T + 2.75e10T^{2} \)
37 \( 1 - 6.08e5T + 9.49e10T^{2} \)
41 \( 1 - 7.87e4T + 1.94e11T^{2} \)
43 \( 1 + 1.05e5T + 2.71e11T^{2} \)
47 \( 1 + 3.88e5T + 5.06e11T^{2} \)
53 \( 1 + 5.39e4T + 1.17e12T^{2} \)
59 \( 1 + 2.74e6T + 2.48e12T^{2} \)
61 \( 1 + 1.42e6T + 3.14e12T^{2} \)
67 \( 1 - 1.87e6T + 6.06e12T^{2} \)
71 \( 1 - 1.52e6T + 9.09e12T^{2} \)
73 \( 1 - 4.26e6T + 1.10e13T^{2} \)
79 \( 1 - 2.27e6T + 1.92e13T^{2} \)
83 \( 1 + 5.13e6T + 2.71e13T^{2} \)
89 \( 1 - 1.00e7T + 4.42e13T^{2} \)
97 \( 1 - 1.40e7T + 8.07e13T^{2} \)
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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

−9.537963825042149203873121619427, −9.337356015630320384458320881283, −7.65227688597312763476122488977, −6.46949231726278051542160700658, −6.10846872415068534220540638425, −5.00320690622622850782710730856, −3.71796038479912250217904216695, −2.43519852905487579036964026250, −1.28499226627412912629034395036, 0, 1.28499226627412912629034395036, 2.43519852905487579036964026250, 3.71796038479912250217904216695, 5.00320690622622850782710730856, 6.10846872415068534220540638425, 6.46949231726278051542160700658, 7.65227688597312763476122488977, 9.337356015630320384458320881283, 9.537963825042149203873121619427

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