Properties

Degree $2$
Conductor $384$
Sign $0.996 + 0.0821i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.246 − 2.98i)3-s + 6.63i·5-s + 0.578·7-s + (−8.87 − 1.47i)9-s − 8.68i·11-s + 17.9·13-s + (19.8 + 1.63i)15-s + 19.0i·17-s + 32.1·19-s + (0.142 − 1.72i)21-s + 20.4i·23-s − 19.0·25-s + (−6.59 + 26.1i)27-s − 22.0i·29-s + 26.2·31-s + ⋯
L(s)  = 1  + (0.0821 − 0.996i)3-s + 1.32i·5-s + 0.0825·7-s + (−0.986 − 0.163i)9-s − 0.789i·11-s + 1.37·13-s + (1.32 + 0.109i)15-s + 1.11i·17-s + 1.69·19-s + (0.00678 − 0.0823i)21-s + 0.890i·23-s − 0.761·25-s + (−0.244 + 0.969i)27-s − 0.759i·29-s + 0.846·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0821i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.996 + 0.0821i$
Motivic weight: \(2\)
Character: $\chi_{384} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 0.996 + 0.0821i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.82091 - 0.0749538i\)
\(L(\frac12)\) \(\approx\) \(1.82091 - 0.0749538i\)
\(L(2)\) 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 \( 1 + (-0.246 + 2.98i)T \)
good5 \( 1 - 6.63iT - 25T^{2} \)
7 \( 1 - 0.578T + 49T^{2} \)
11 \( 1 + 8.68iT - 121T^{2} \)
13 \( 1 - 17.9T + 169T^{2} \)
17 \( 1 - 19.0iT - 289T^{2} \)
19 \( 1 - 32.1T + 361T^{2} \)
23 \( 1 - 20.4iT - 529T^{2} \)
29 \( 1 + 22.0iT - 841T^{2} \)
31 \( 1 - 26.2T + 961T^{2} \)
37 \( 1 - 53.3T + 1.36e3T^{2} \)
41 \( 1 + 35.6iT - 1.68e3T^{2} \)
43 \( 1 - 50.4T + 1.84e3T^{2} \)
47 \( 1 + 30.6iT - 2.20e3T^{2} \)
53 \( 1 - 88.8iT - 2.80e3T^{2} \)
59 \( 1 + 63.1iT - 3.48e3T^{2} \)
61 \( 1 + 33.5T + 3.72e3T^{2} \)
67 \( 1 + 108.T + 4.48e3T^{2} \)
71 \( 1 - 59.3iT - 5.04e3T^{2} \)
73 \( 1 + 5.60T + 5.32e3T^{2} \)
79 \( 1 + 78.9T + 6.24e3T^{2} \)
83 \( 1 + 48.5iT - 6.88e3T^{2} \)
89 \( 1 - 58.7iT - 7.92e3T^{2} \)
97 \( 1 - 93.3T + 9.40e3T^{2} \)
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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

−11.25489753963352681016935165481, −10.45550602362944871373129619386, −9.156624487989610379511090514098, −8.067645744099440995520089890556, −7.40896722265235815331968313165, −6.24060933318572958926851053393, −5.84527256605935110303966202158, −3.66368705816402295080233917897, −2.81121875331713600259995411607, −1.22875853081949797982988819483, 1.03880484066540849245883298221, 3.03050922531264360173349890288, 4.41764685613205876467269186464, 4.97800081225600688712124014947, 6.05900692232077575161043915605, 7.64120792319083874926343585574, 8.632794861916531336186608050988, 9.300236578237207432860559159342, 9.998406078665805110679023147935, 11.20061092329133965436149638848

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