Properties

Degree $2$
Conductor $384$
Sign $-0.478 - 0.877i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 1.22i)3-s + (−0.909 − 0.909i)5-s + 0.654·7-s + 2.99i·9-s + (−13.3 + 13.3i)11-s + (−8.32 + 8.32i)13-s − 2.22i·15-s − 3.93·17-s + (16.8 + 16.8i)19-s + (0.801 + 0.801i)21-s + 23.1·23-s − 23.3i·25-s + (−3.67 + 3.67i)27-s + (−35.6 + 35.6i)29-s + 45.5i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.181 − 0.181i)5-s + 0.0935·7-s + 0.333i·9-s + (−1.21 + 1.21i)11-s + (−0.640 + 0.640i)13-s − 0.148i·15-s − 0.231·17-s + (0.889 + 0.889i)19-s + (0.0381 + 0.0381i)21-s + 1.00·23-s − 0.933i·25-s + (−0.136 + 0.136i)27-s + (−1.22 + 1.22i)29-s + 1.46i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.877i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.478 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.478 - 0.877i$
Motivic weight: \(2\)
Character: $\chi_{384} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.478 - 0.877i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.643803 + 1.08451i\)
\(L(\frac12)\) \(\approx\) \(0.643803 + 1.08451i\)
\(L(2)\) 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 \( 1 + (-1.22 - 1.22i)T \)
good5 \( 1 + (0.909 + 0.909i)T + 25iT^{2} \)
7 \( 1 - 0.654T + 49T^{2} \)
11 \( 1 + (13.3 - 13.3i)T - 121iT^{2} \)
13 \( 1 + (8.32 - 8.32i)T - 169iT^{2} \)
17 \( 1 + 3.93T + 289T^{2} \)
19 \( 1 + (-16.8 - 16.8i)T + 361iT^{2} \)
23 \( 1 - 23.1T + 529T^{2} \)
29 \( 1 + (35.6 - 35.6i)T - 841iT^{2} \)
31 \( 1 - 45.5iT - 961T^{2} \)
37 \( 1 + (10.1 + 10.1i)T + 1.36e3iT^{2} \)
41 \( 1 - 28.4iT - 1.68e3T^{2} \)
43 \( 1 + (-22.7 + 22.7i)T - 1.84e3iT^{2} \)
47 \( 1 - 10.7iT - 2.20e3T^{2} \)
53 \( 1 + (41.5 + 41.5i)T + 2.80e3iT^{2} \)
59 \( 1 + (21.0 - 21.0i)T - 3.48e3iT^{2} \)
61 \( 1 + (-68.7 + 68.7i)T - 3.72e3iT^{2} \)
67 \( 1 + (-67.8 - 67.8i)T + 4.48e3iT^{2} \)
71 \( 1 + 33.3T + 5.04e3T^{2} \)
73 \( 1 - 18.6iT - 5.32e3T^{2} \)
79 \( 1 - 6.29iT - 6.24e3T^{2} \)
83 \( 1 + (72.0 + 72.0i)T + 6.88e3iT^{2} \)
89 \( 1 + 10.6iT - 7.92e3T^{2} \)
97 \( 1 - 143.T + 9.40e3T^{2} \)
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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

−11.34922878718695571600481347649, −10.33707010166980717149314287120, −9.678702977179352473170976237266, −8.714953389378009082677522998383, −7.67772102079337070481142869040, −6.96629937841019621803567533903, −5.26683323847084034992940783171, −4.63872652127383658092287268268, −3.24801682127452883812333213725, −1.92749674859493004365827000791, 0.50927319740506714287845970381, 2.50117063737815374756563219969, 3.39418632933417042811343314967, 5.03672262559976724083975946007, 5.94885268927827000868709795992, 7.38371183898461242361493172090, 7.82017156705091404212516968075, 8.926302047568020728472746945652, 9.832795010455838238556873702395, 11.05135169229733117955139120040

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