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Properties

Label 2-384-24.11-c3-0-19
Degree $2$
Conductor $384$
Sign $0.829 - 0.557i$
Analytic cond. $22.6567$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.09 − i)3-s + 10.1·5-s + 10.1i·7-s + (24.9 + 10.1i)9-s − 46i·11-s + 44i·13-s + (−51.9 − 10.1i)15-s + 20.3i·17-s + 10.1·19-s + (10.1 − 51.9i)21-s − 88·23-s − 21.0·25-s + (−117. − 76.9i)27-s + 254.·29-s + 214. i·31-s + ⋯
L(s)  = 1  + (−0.981 − 0.192i)3-s + 0.912·5-s + 0.550i·7-s + (0.925 + 0.377i)9-s − 1.26i·11-s + 0.938i·13-s + (−0.895 − 0.175i)15-s + 0.290i·17-s + 0.123·19-s + (0.105 − 0.540i)21-s − 0.797·23-s − 0.168·25-s + (−0.835 − 0.548i)27-s + 1.63·29-s + 1.24i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.557i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.829 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.829 - 0.557i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 0.829 - 0.557i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.527169044\)
\(L(\frac12)\) \(\approx\) \(1.527169044\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.09 + i)T \)
good5 \( 1 - 10.1T + 125T^{2} \)
7 \( 1 - 10.1iT - 343T^{2} \)
11 \( 1 + 46iT - 1.33e3T^{2} \)
13 \( 1 - 44iT - 2.19e3T^{2} \)
17 \( 1 - 20.3iT - 4.91e3T^{2} \)
19 \( 1 - 10.1T + 6.85e3T^{2} \)
23 \( 1 + 88T + 1.21e4T^{2} \)
29 \( 1 - 254.T + 2.43e4T^{2} \)
31 \( 1 - 214. iT - 2.97e4T^{2} \)
37 \( 1 + 332iT - 5.06e4T^{2} \)
41 \( 1 - 489. iT - 6.89e4T^{2} \)
43 \( 1 - 234.T + 7.95e4T^{2} \)
47 \( 1 - 384T + 1.03e5T^{2} \)
53 \( 1 - 458.T + 1.48e5T^{2} \)
59 \( 1 - 630iT - 2.05e5T^{2} \)
61 \( 1 - 236iT - 2.26e5T^{2} \)
67 \( 1 + 50.9T + 3.00e5T^{2} \)
71 \( 1 - 680T + 3.57e5T^{2} \)
73 \( 1 - 422T + 3.89e5T^{2} \)
79 \( 1 - 744. iT - 4.93e5T^{2} \)
83 \( 1 - 186iT - 5.71e5T^{2} \)
89 \( 1 - 958. iT - 7.04e5T^{2} \)
97 \( 1 + 1.06e3T + 9.12e5T^{2} \)
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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

−11.00928378913770572667726907955, −10.23034102794815848653572671845, −9.253472126991783162007670203221, −8.309779273851233495428399830076, −6.91180106037657762344760500406, −6.02555236952748195748887724984, −5.51800786662508369692804379637, −4.18276300448995929055912623338, −2.43259681218300691228305394814, −1.09203255405296752864955148613, 0.71936775367820822958079707610, 2.20088709861418325556210129523, 4.03328660146504025408164781197, 5.05798927625358856048916462531, 5.94607300455247912007257813586, 6.89008274989190101928214915304, 7.82777149077445649775703971754, 9.376160248017737515105968592255, 10.17540224364110144499451446293, 10.49205336930054759399227065047

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