Properties

Label 2-384-3.2-c2-0-12
Degree $2$
Conductor $384$
Sign $0.442 - 0.896i$
Analytic cond. $10.4632$
Root an. cond. $3.23469$
Motivic weight $2$
Arithmetic yes
Rational no
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: $0.442 - 0.896i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 0.442 - 0.896i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.00038 + 1.24359i\)
\(L(\frac12)\) \(\approx\) \(2.00038 + 1.24359i\)
\(L(2)\) 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

−10.99689260678798524602447144080, −10.33788258856438132672243452206, −9.411349989541419422721403150707, −8.589095580498849476982579243694, −7.63769048206776040205521456717, −6.79234979990685284082896741918, −5.14798194466945600148460143175, −4.31706384912619951346323518839, −3.03074045857510159345439850913, −1.78971122078102450730894375999, 1.03453143870669941120534814728, 2.58937487059671341088290038810, 3.65344511831264177838484568559, 5.05272162940116153190431927820, 6.26021929297287542581539893143, 7.52800056249670874774621814691, 7.997744306463461348559513379156, 9.210713390941147470972317155430, 9.675474251168946409471505379977, 11.18126769190662395472055903590

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