Properties

Label 2-384-3.2-c4-0-15
Degree $2$
Conductor $384$
Sign $-0.891 + 0.453i$
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  + (4.08 + 8.01i)3-s + 47.0i·5-s + 13.4·7-s + (−47.6 + 65.5i)9-s − 135. i·11-s − 80.8·13-s + (−377. + 192. i)15-s + 430. i·17-s − 43.5·19-s + (54.9 + 107. i)21-s + 873. i·23-s − 1.59e3·25-s + (−719. − 114. i)27-s − 707. i·29-s + 677.·31-s + ⋯
L(s)  = 1  + (0.453 + 0.891i)3-s + 1.88i·5-s + 0.274·7-s + (−0.587 + 0.808i)9-s − 1.11i·11-s − 0.478·13-s + (−1.67 + 0.854i)15-s + 1.49i·17-s − 0.120·19-s + (0.124 + 0.244i)21-s + 1.65i·23-s − 2.54·25-s + (−0.987 − 0.156i)27-s − 0.840i·29-s + 0.705·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.891 + 0.453i$
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: \((2,\ 384,\ (\ :2),\ -0.891 + 0.453i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.537940490\)
\(L(\frac12)\) \(\approx\) \(1.537940490\)
\(L(3)\) 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 \( 1 + (-4.08 - 8.01i)T \)
good5 \( 1 - 47.0iT - 625T^{2} \)
7 \( 1 - 13.4T + 2.40e3T^{2} \)
11 \( 1 + 135. iT - 1.46e4T^{2} \)
13 \( 1 + 80.8T + 2.85e4T^{2} \)
17 \( 1 - 430. iT - 8.35e4T^{2} \)
19 \( 1 + 43.5T + 1.30e5T^{2} \)
23 \( 1 - 873. iT - 2.79e5T^{2} \)
29 \( 1 + 707. iT - 7.07e5T^{2} \)
31 \( 1 - 677.T + 9.23e5T^{2} \)
37 \( 1 - 1.63e3T + 1.87e6T^{2} \)
41 \( 1 + 1.22e3iT - 2.82e6T^{2} \)
43 \( 1 + 1.43e3T + 3.41e6T^{2} \)
47 \( 1 + 2.06e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.46e3iT - 7.89e6T^{2} \)
59 \( 1 + 470. iT - 1.21e7T^{2} \)
61 \( 1 + 39.9T + 1.38e7T^{2} \)
67 \( 1 - 4.52e3T + 2.01e7T^{2} \)
71 \( 1 - 9.85e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.27e3T + 2.83e7T^{2} \)
79 \( 1 - 6.93e3T + 3.89e7T^{2} \)
83 \( 1 + 1.27e4iT - 4.74e7T^{2} \)
89 \( 1 - 9.78e3iT - 6.27e7T^{2} \)
97 \( 1 + 9.32e3T + 8.85e7T^{2} \)
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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

−11.15375035299656877029616514985, −10.29791969667859373217323045488, −9.723622674805926597969806814534, −8.372046268028204994163306373642, −7.66549606205066266625989986340, −6.42115488499179082405012822944, −5.55832139721510692709064046271, −3.92726797305344805878189639245, −3.25354840009936477173801134772, −2.18216816317090146436935925117, 0.40158405848014785282479756981, 1.41963700355927580300746603216, 2.58340127032561573640364065575, 4.48914950197873528244113195884, 5.03425095800699122626469277667, 6.47051087662206637987488721581, 7.58341473896419752409732953913, 8.291660805348687748614919420580, 9.171049290774593672266010918611, 9.780049953932637177808533559591

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