Properties

Degree $2$
Conductor $384$
Sign $0.647 - 0.761i$
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 + (−1.00 + 1.00i)5-s + 10.0·7-s − 2.99i·9-s + (−2.26 − 2.26i)11-s + (6.88 + 6.88i)13-s − 2.46i·15-s − 22.3·17-s + (16.8 − 16.8i)19-s + (−12.2 + 12.2i)21-s + 33.2·23-s + 22.9i·25-s + (3.67 + 3.67i)27-s + (24.6 + 24.6i)29-s + 41.3i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.201 + 0.201i)5-s + 1.43·7-s − 0.333i·9-s + (−0.205 − 0.205i)11-s + (0.529 + 0.529i)13-s − 0.164i·15-s − 1.31·17-s + (0.889 − 0.889i)19-s + (−0.584 + 0.584i)21-s + 1.44·23-s + 0.918i·25-s + (0.136 + 0.136i)27-s + (0.849 + 0.849i)29-s + 1.33i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 - 0.761i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.647 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.647 - 0.761i$
Motivic weight: \(2\)
Character: $\chi_{384} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 0.647 - 0.761i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.48809 + 0.687897i\)
\(L(\frac12)\) \(\approx\) \(1.48809 + 0.687897i\)
\(L(2)\) 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 \( 1 + (1.00 - 1.00i)T - 25iT^{2} \)
7 \( 1 - 10.0T + 49T^{2} \)
11 \( 1 + (2.26 + 2.26i)T + 121iT^{2} \)
13 \( 1 + (-6.88 - 6.88i)T + 169iT^{2} \)
17 \( 1 + 22.3T + 289T^{2} \)
19 \( 1 + (-16.8 + 16.8i)T - 361iT^{2} \)
23 \( 1 - 33.2T + 529T^{2} \)
29 \( 1 + (-24.6 - 24.6i)T + 841iT^{2} \)
31 \( 1 - 41.3iT - 961T^{2} \)
37 \( 1 + (-6.60 + 6.60i)T - 1.36e3iT^{2} \)
41 \( 1 - 47.1iT - 1.68e3T^{2} \)
43 \( 1 + (-48.8 - 48.8i)T + 1.84e3iT^{2} \)
47 \( 1 + 45.6iT - 2.20e3T^{2} \)
53 \( 1 + (25.1 - 25.1i)T - 2.80e3iT^{2} \)
59 \( 1 + (6.23 + 6.23i)T + 3.48e3iT^{2} \)
61 \( 1 + (35.9 + 35.9i)T + 3.72e3iT^{2} \)
67 \( 1 + (10.2 - 10.2i)T - 4.48e3iT^{2} \)
71 \( 1 - 11.9T + 5.04e3T^{2} \)
73 \( 1 + 111. iT - 5.32e3T^{2} \)
79 \( 1 + 4.46iT - 6.24e3T^{2} \)
83 \( 1 + (10.1 - 10.1i)T - 6.88e3iT^{2} \)
89 \( 1 - 21.9iT - 7.92e3T^{2} \)
97 \( 1 - 107.T + 9.40e3T^{2} \)
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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

−11.02950016946582805627547501761, −10.85856257284395915734412963647, −9.250706979961288020867107295656, −8.645143021362120055680411878205, −7.44852589410132976432634870052, −6.54126603219156429073904603530, −5.09234563022895020890357635113, −4.60350596304135320943164906672, −3.06923525630409796623983440158, −1.31297754247809624792285302133, 0.930583109503621570892902515107, 2.34274285789178043795997409329, 4.20368966311082453632500186066, 5.10889608294749557469416696978, 6.12753992369975121569697094403, 7.40794016004736130066260323082, 8.091925609221694358577036060766, 8.958155663180588656811484642019, 10.35391556204361238216832702329, 11.13964384417941734021181149511

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