Properties

Label 2-384-3.2-c4-0-1
Degree $2$
Conductor $384$
Sign $-0.683 + 0.730i$
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  + (6.57 + 6.14i)3-s + 9.58i·5-s − 22.6·7-s + (5.37 + 80.8i)9-s − 2.72i·11-s − 220.·13-s + (−58.9 + 63.0i)15-s − 172. i·17-s − 275.·19-s + (−149. − 139. i)21-s − 420. i·23-s + 533.·25-s + (−461. + 564. i)27-s − 207. i·29-s − 1.08e3·31-s + ⋯
L(s)  = 1  + (0.730 + 0.683i)3-s + 0.383i·5-s − 0.462·7-s + (0.0663 + 0.997i)9-s − 0.0224i·11-s − 1.30·13-s + (−0.262 + 0.280i)15-s − 0.598i·17-s − 0.763·19-s + (−0.337 − 0.316i)21-s − 0.794i·23-s + 0.852·25-s + (−0.633 + 0.773i)27-s − 0.246i·29-s − 1.13·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.683 + 0.730i$
Analytic conductor: \(39.6940\)
Root analytic conductor: \(6.30032\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :2),\ -0.683 + 0.730i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.06053763825\)
\(L(\frac12)\) \(\approx\) \(0.06053763825\)
\(L(3)\) 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 \( 1 + (-6.57 - 6.14i)T \)
good5 \( 1 - 9.58iT - 625T^{2} \)
7 \( 1 + 22.6T + 2.40e3T^{2} \)
11 \( 1 + 2.72iT - 1.46e4T^{2} \)
13 \( 1 + 220.T + 2.85e4T^{2} \)
17 \( 1 + 172. iT - 8.35e4T^{2} \)
19 \( 1 + 275.T + 1.30e5T^{2} \)
23 \( 1 + 420. iT - 2.79e5T^{2} \)
29 \( 1 + 207. iT - 7.07e5T^{2} \)
31 \( 1 + 1.08e3T + 9.23e5T^{2} \)
37 \( 1 + 1.26e3T + 1.87e6T^{2} \)
41 \( 1 + 1.23e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.38e3T + 3.41e6T^{2} \)
47 \( 1 - 1.43e3iT - 4.87e6T^{2} \)
53 \( 1 + 3.63e3iT - 7.89e6T^{2} \)
59 \( 1 - 5.98e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.17e3T + 1.38e7T^{2} \)
67 \( 1 - 5.98e3T + 2.01e7T^{2} \)
71 \( 1 + 7.55e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.32e3T + 2.83e7T^{2} \)
79 \( 1 + 279.T + 3.89e7T^{2} \)
83 \( 1 - 4.33e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.25e4iT - 6.27e7T^{2} \)
97 \( 1 + 1.65e4T + 8.85e7T^{2} \)
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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

−10.98641199297128391586312959588, −10.30147555398716407191278721659, −9.472611316343708447118687356500, −8.711790170289734939239104148097, −7.57086648176846723235095972753, −6.74552294075573726925379538973, −5.29185905315381200002062009579, −4.31261422319690688761361821377, −3.10636078383973240934170243329, −2.22936180719508515386425263104, 0.01426244226729586722237205582, 1.55796590773195879480148859158, 2.73115910812982909319393413833, 3.90891046067411554837731863077, 5.25705126225992689065521420448, 6.53231957454063805701800218971, 7.32936196611337413828639699910, 8.278569008010120344363840919570, 9.161982297897429299138603448424, 9.889260787092964058130541793540

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