Properties

Label 2-384-8.5-c3-0-11
Degree $2$
Conductor $384$
Sign $1$
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  + 3i·3-s − 5.67i·5-s − 33.0·7-s − 9·9-s − 34.6i·11-s + 82.2i·13-s + 17.0·15-s + 97.8·17-s − 55.8i·19-s − 99.2i·21-s + 130.·23-s + 92.7·25-s − 27i·27-s − 147. i·29-s + 101.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.507i·5-s − 1.78·7-s − 0.333·9-s − 0.949i·11-s + 1.75i·13-s + 0.293·15-s + 1.39·17-s − 0.674i·19-s − 1.03i·21-s + 1.18·23-s + 0.742·25-s − 0.192i·27-s − 0.944i·29-s + 0.586·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $1$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.433652702\)
\(L(\frac12)\) \(\approx\) \(1.433652702\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
good5 \( 1 + 5.67iT - 125T^{2} \)
7 \( 1 + 33.0T + 343T^{2} \)
11 \( 1 + 34.6iT - 1.33e3T^{2} \)
13 \( 1 - 82.2iT - 2.19e3T^{2} \)
17 \( 1 - 97.8T + 4.91e3T^{2} \)
19 \( 1 + 55.8iT - 6.85e3T^{2} \)
23 \( 1 - 130.T + 1.21e4T^{2} \)
29 \( 1 + 147. iT - 2.43e4T^{2} \)
31 \( 1 - 101.T + 2.97e4T^{2} \)
37 \( 1 - 184. iT - 5.06e4T^{2} \)
41 \( 1 - 237.T + 6.89e4T^{2} \)
43 \( 1 + 199. iT - 7.95e4T^{2} \)
47 \( 1 + 334.T + 1.03e5T^{2} \)
53 \( 1 - 102. iT - 1.48e5T^{2} \)
59 \( 1 + 105. iT - 2.05e5T^{2} \)
61 \( 1 + 717. iT - 2.26e5T^{2} \)
67 \( 1 - 316. iT - 3.00e5T^{2} \)
71 \( 1 - 800.T + 3.57e5T^{2} \)
73 \( 1 - 301.T + 3.89e5T^{2} \)
79 \( 1 + 42.8T + 4.93e5T^{2} \)
83 \( 1 - 1.23e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.32e3T + 7.04e5T^{2} \)
97 \( 1 + 505.T + 9.12e5T^{2} \)
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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

−10.88299188189371940187731900320, −9.675399047317885029930647442758, −9.357743104150149974562932895015, −8.413690259444698128281404420713, −6.89733677808812875822219617684, −6.18283851871160637158265817104, −4.99202522239087900764308930077, −3.76922321981611224359324571795, −2.86935578336640721240699483417, −0.72159107310488193175810629937, 0.853435154646252357519252737931, 2.83048369814012209256059605780, 3.39766114554702844661397290257, 5.30899426695861578984593132780, 6.25747037976595670201122862147, 7.12277264300798130602130037961, 7.87297816268360121848597476719, 9.231729314123107206306254359233, 10.10640124676645591232494209000, 10.64728920189489479219659430179

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