Properties

Degree $2$
Conductor $384$
Sign $0.680 - 0.732i$
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 + (5.24 + 5.24i)5-s + 5.32·7-s + 2.99i·9-s + (12.2 − 12.2i)11-s + (5.73 − 5.73i)13-s + 12.8i·15-s − 23.3·17-s + (11.7 + 11.7i)19-s + (6.52 + 6.52i)21-s − 5.80·23-s + 29.9i·25-s + (−3.67 + 3.67i)27-s + (−18.3 + 18.3i)29-s − 16.9i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (1.04 + 1.04i)5-s + 0.761·7-s + 0.333i·9-s + (1.11 − 1.11i)11-s + (0.441 − 0.441i)13-s + 0.856i·15-s − 1.37·17-s + (0.618 + 0.618i)19-s + (0.310 + 0.310i)21-s − 0.252·23-s + 1.19i·25-s + (−0.136 + 0.136i)27-s + (−0.634 + 0.634i)29-s − 0.545i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.680 - 0.732i$
Motivic weight: \(2\)
Character: $\chi_{384} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 0.680 - 0.732i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.37354 + 1.03456i\)
\(L(\frac12)\) \(\approx\) \(2.37354 + 1.03456i\)
\(L(2)\) 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 \( 1 + (-5.24 - 5.24i)T + 25iT^{2} \)
7 \( 1 - 5.32T + 49T^{2} \)
11 \( 1 + (-12.2 + 12.2i)T - 121iT^{2} \)
13 \( 1 + (-5.73 + 5.73i)T - 169iT^{2} \)
17 \( 1 + 23.3T + 289T^{2} \)
19 \( 1 + (-11.7 - 11.7i)T + 361iT^{2} \)
23 \( 1 + 5.80T + 529T^{2} \)
29 \( 1 + (18.3 - 18.3i)T - 841iT^{2} \)
31 \( 1 + 16.9iT - 961T^{2} \)
37 \( 1 + (15.3 + 15.3i)T + 1.36e3iT^{2} \)
41 \( 1 - 29.2iT - 1.68e3T^{2} \)
43 \( 1 + (-33.4 + 33.4i)T - 1.84e3iT^{2} \)
47 \( 1 + 18.2iT - 2.20e3T^{2} \)
53 \( 1 + (-66.9 - 66.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (27.1 - 27.1i)T - 3.48e3iT^{2} \)
61 \( 1 + (65.2 - 65.2i)T - 3.72e3iT^{2} \)
67 \( 1 + (37.6 + 37.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 42.6T + 5.04e3T^{2} \)
73 \( 1 + 106. iT - 5.32e3T^{2} \)
79 \( 1 + 21.2iT - 6.24e3T^{2} \)
83 \( 1 + (-24.1 - 24.1i)T + 6.88e3iT^{2} \)
89 \( 1 + 52.8iT - 7.92e3T^{2} \)
97 \( 1 + 21.0T + 9.40e3T^{2} \)
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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

−10.94492043691830808314352929525, −10.53043440361281508645923877263, −9.317181143497099534917229784104, −8.715722820320391224368122151776, −7.49709363051240945994404931566, −6.33449001244123784161003803649, −5.61783675678265225288135876890, −4.08738266095117036798056012227, −2.97628138957115211081011557826, −1.67959182624567249717342122384, 1.36971579944281000441713733630, 2.12559985617973425805000920433, 4.17948966558699523179805271526, 5.02269940755024962817650319814, 6.28687988742998570077593464955, 7.18745626296951475454854459657, 8.470710388007541303750861868248, 9.152401541011923594766492580272, 9.718430581632679017347835445430, 11.15297521087253165821539422825

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