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Properties

Label 2-384-8.5-c9-0-58
Degree $2$
Conductor $384$
Sign $-0.707 + 0.707i$
Analytic cond. $197.773$
Root an. cond. $14.0632$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 81i·3-s − 1.78e3i·5-s + 4.96e3·7-s − 6.56e3·9-s + 5.37e4i·11-s + 5.42e4i·13-s + 1.44e5·15-s − 5.54e5·17-s + 6.50e5i·19-s + 4.02e5i·21-s − 1.83e5·23-s − 1.22e6·25-s − 5.31e5i·27-s − 2.02e6i·29-s + 3.43e6·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.27i·5-s + 0.781·7-s − 0.333·9-s + 1.10i·11-s + 0.526i·13-s + 0.736·15-s − 1.61·17-s + 1.14i·19-s + 0.451i·21-s − 0.136·23-s − 0.628·25-s − 0.192i·27-s − 0.531i·29-s + 0.667·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(197.773\)
Root analytic conductor: \(14.0632\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :9/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.5102117766\)
\(L(\frac12)\) \(\approx\) \(0.5102117766\)
\(L(\frac{11}{2})\) 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 \( 1 - 81iT \)
good5 \( 1 + 1.78e3iT - 1.95e6T^{2} \)
7 \( 1 - 4.96e3T + 4.03e7T^{2} \)
11 \( 1 - 5.37e4iT - 2.35e9T^{2} \)
13 \( 1 - 5.42e4iT - 1.06e10T^{2} \)
17 \( 1 + 5.54e5T + 1.18e11T^{2} \)
19 \( 1 - 6.50e5iT - 3.22e11T^{2} \)
23 \( 1 + 1.83e5T + 1.80e12T^{2} \)
29 \( 1 + 2.02e6iT - 1.45e13T^{2} \)
31 \( 1 - 3.43e6T + 2.64e13T^{2} \)
37 \( 1 - 1.30e6iT - 1.29e14T^{2} \)
41 \( 1 - 1.57e5T + 3.27e14T^{2} \)
43 \( 1 + 2.10e6iT - 5.02e14T^{2} \)
47 \( 1 - 4.73e7T + 1.11e15T^{2} \)
53 \( 1 + 1.00e8iT - 3.29e15T^{2} \)
59 \( 1 + 1.80e7iT - 8.66e15T^{2} \)
61 \( 1 - 5.48e7iT - 1.16e16T^{2} \)
67 \( 1 + 2.13e8iT - 2.72e16T^{2} \)
71 \( 1 - 1.64e8T + 4.58e16T^{2} \)
73 \( 1 + 7.58e7T + 5.88e16T^{2} \)
79 \( 1 + 4.25e8T + 1.19e17T^{2} \)
83 \( 1 + 1.06e8iT - 1.86e17T^{2} \)
89 \( 1 + 1.02e9T + 3.50e17T^{2} \)
97 \( 1 - 6.24e8T + 7.60e17T^{2} \)
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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

−9.394782890525243189840238536534, −8.644425739481099729164722220889, −7.892283315818834978316543135716, −6.61669448857337676703854017629, −5.35004686740485941590968587408, −4.55372468761674789107729304927, −4.08195417148493995528770448294, −2.23501794170738129759464609022, −1.43936647983966244321785634031, −0.094361886474559750799705281455, 1.04291844149129832418222866307, 2.39329298789251291737317427228, 2.99271533915265434366917361266, 4.36054738038520666202795540966, 5.63299555897300501039036802395, 6.57942930922355830307813028311, 7.24837799969424103094384819302, 8.273977175684807732198782638612, 9.035153932060024957659613001578, 10.53244790449334472225326188333

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