Properties

Degree $2$
Conductor $384$
Sign $-0.442 + 0.896i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.69 − 1.32i)3-s − 0.640i·5-s + 2.72·7-s + (5.47 + 7.14i)9-s − 11.2i·11-s + 5.25·13-s + (−0.849 + 1.72i)15-s + 14.8i·17-s − 15.0·19-s + (−7.31 − 3.61i)21-s − 36.4i·23-s + 24.5·25-s + (−5.24 − 26.4i)27-s − 51.7i·29-s − 36.5·31-s + ⋯
L(s)  = 1  + (−0.896 − 0.442i)3-s − 0.128i·5-s + 0.388·7-s + (0.608 + 0.793i)9-s − 1.02i·11-s + 0.403·13-s + (−0.0566 + 0.114i)15-s + 0.874i·17-s − 0.793·19-s + (−0.348 − 0.171i)21-s − 1.58i·23-s + 0.983·25-s + (−0.194 − 0.980i)27-s − 1.78i·29-s − 1.17·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.442 + 0.896i$
Motivic weight: \(2\)
Character: $\chi_{384} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.442 + 0.896i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.499103 - 0.802834i\)
\(L(\frac12)\) \(\approx\) \(0.499103 - 0.802834i\)
\(L(2)\) 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 \( 1 + (2.69 + 1.32i)T \)
good5 \( 1 + 0.640iT - 25T^{2} \)
7 \( 1 - 2.72T + 49T^{2} \)
11 \( 1 + 11.2iT - 121T^{2} \)
13 \( 1 - 5.25T + 169T^{2} \)
17 \( 1 - 14.8iT - 289T^{2} \)
19 \( 1 + 15.0T + 361T^{2} \)
23 \( 1 + 36.4iT - 529T^{2} \)
29 \( 1 + 51.7iT - 841T^{2} \)
31 \( 1 + 36.5T + 961T^{2} \)
37 \( 1 + 63.6T + 1.36e3T^{2} \)
41 \( 1 + 12.1iT - 1.68e3T^{2} \)
43 \( 1 - 11.8T + 1.84e3T^{2} \)
47 \( 1 + 61.1iT - 2.20e3T^{2} \)
53 \( 1 + 59.1iT - 2.80e3T^{2} \)
59 \( 1 - 37.2iT - 3.48e3T^{2} \)
61 \( 1 - 58.1T + 3.72e3T^{2} \)
67 \( 1 + 23.0T + 4.48e3T^{2} \)
71 \( 1 + 7.29iT - 5.04e3T^{2} \)
73 \( 1 - 73.4T + 5.32e3T^{2} \)
79 \( 1 - 58.5T + 6.24e3T^{2} \)
83 \( 1 - 32.3iT - 6.88e3T^{2} \)
89 \( 1 + 112. iT - 7.92e3T^{2} \)
97 \( 1 + 80.0T + 9.40e3T^{2} \)
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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

−10.84539149085857233823594377404, −10.32775104834531315627073163006, −8.711561218698125489855632845947, −8.145804381926795255483915880162, −6.80517383674973153906727399690, −6.07058857343525231746846708977, −5.08081236295656852466333229404, −3.88581979304772611454258210631, −2.03134800341827567272880778125, −0.48676195618193759346894659144, 1.53439496503711853205364865628, 3.45917682087197897100506223708, 4.72481414625756167809110123867, 5.41740163988124002706256217648, 6.72791455764163660877218915986, 7.40983558433932054798411590595, 8.898853412798002992134083899447, 9.662749180374282618687375302426, 10.73094436070459936891783259488, 11.20215321420023982332472436814

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