Properties

Degree $2$
Conductor $384$
Sign $-0.877 - 0.478i$
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

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

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.877 - 0.478i$
Motivic weight: \(2\)
Character: $\chi_{384} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.877 - 0.478i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.172029 + 0.674636i\)
\(L(\frac12)\) \(\approx\) \(0.172029 + 0.674636i\)
\(L(2)\) 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

−11.62137534589995847617127624761, −10.44059560197233481666487545836, −9.852207315806563163230983280034, −8.915502645741784333358702915394, −7.66163288490088483142592730093, −6.76285774862165210224644961735, −5.73196574752938207719967382053, −4.51669074682780351147051573015, −3.62986643716635862121576511947, −1.85576428378718089614376426251, 0.31332792435440780675867668442, 1.99396129439027870398219486126, 3.68354576329253725441661883641, 4.79355877531705059144422636548, 6.15727150728932122066302746707, 6.71320129003214800138087834410, 7.966451471711764231817607492918, 8.877231494649443427226968747991, 9.739225938090997384411566032622, 11.11726288409184789654187073693

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