Properties

Label 2-384-24.5-c4-0-39
Degree $2$
Conductor $384$
Sign $0.333 + 0.942i$
Analytic cond. $39.6940$
Root an. cond. $6.30032$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3 + 8.48i)3-s − 37.9·5-s + 37.9·7-s + (−62.9 + 50.9i)9-s − 134·11-s + 321. i·13-s + (−113. − 321. i)15-s − 237. i·17-s − 492. i·19-s + (113. + 321. i)21-s + 643. i·23-s + 815·25-s + (−620. − 381. i)27-s − 796.·29-s + 1.10e3·31-s + ⋯
L(s)  = 1  + (0.333 + 0.942i)3-s − 1.51·5-s + 0.774·7-s + (−0.777 + 0.628i)9-s − 1.10·11-s + 1.90i·13-s + (−0.505 − 1.43i)15-s − 0.822i·17-s − 1.36i·19-s + (0.258 + 0.730i)21-s + 1.21i·23-s + 1.30·25-s + (−0.851 − 0.523i)27-s − 0.947·29-s + 1.14·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.333 + 0.942i$
Analytic conductor: \(39.6940\)
Root analytic conductor: \(6.30032\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :2),\ 0.333 + 0.942i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.4470895739\)
\(L(\frac12)\) \(\approx\) \(0.4470895739\)
\(L(3)\) 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 \( 1 + (-3 - 8.48i)T \)
good5 \( 1 + 37.9T + 625T^{2} \)
7 \( 1 - 37.9T + 2.40e3T^{2} \)
11 \( 1 + 134T + 1.46e4T^{2} \)
13 \( 1 - 321. iT - 2.85e4T^{2} \)
17 \( 1 + 237. iT - 8.35e4T^{2} \)
19 \( 1 + 492. iT - 1.30e5T^{2} \)
23 \( 1 - 643. iT - 2.79e5T^{2} \)
29 \( 1 + 796.T + 7.07e5T^{2} \)
31 \( 1 - 1.10e3T + 9.23e5T^{2} \)
37 \( 1 + 1.60e3iT - 1.87e6T^{2} \)
41 \( 1 + 1.28e3iT - 2.82e6T^{2} \)
43 \( 1 - 424. iT - 3.41e6T^{2} \)
47 \( 1 + 2.57e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.63e3T + 7.89e6T^{2} \)
59 \( 1 - 1.38e3T + 1.21e7T^{2} \)
61 \( 1 - 321. iT - 1.38e7T^{2} \)
67 \( 1 + 186. iT - 2.01e7T^{2} \)
71 \( 1 + 1.93e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.89e3T + 2.83e7T^{2} \)
79 \( 1 + 1.47e3T + 3.89e7T^{2} \)
83 \( 1 - 5.91e3T + 4.74e7T^{2} \)
89 \( 1 + 9.06e3iT - 6.27e7T^{2} \)
97 \( 1 - 5.85e3T + 8.85e7T^{2} \)
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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

−10.76483300978353835818409279021, −9.453009859167576410550801797758, −8.742160592726702767474757558586, −7.78332577230208700304679599811, −7.10430480391362138649301682272, −5.19985958850412922096076595441, −4.51948227567807503114102543737, −3.64851983149401973492872741171, −2.31973833912136690193194768659, −0.14272126131083889913452826852, 1.02888819555824534285800994232, 2.66718703119923065581349218555, 3.68890912417119097328602323510, 5.04777850826941420432333828960, 6.20249309461562686878591516958, 7.63010264034747106622935490636, 8.104528871911236134630492685676, 8.302918170253086314552140972825, 10.23536139450935246434569148596, 10.97962159588147097139061176581

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