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: $\chi_{384} (31, \cdot )$
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

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

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