Properties

Label 2-384-24.11-c3-0-3
Degree $2$
Conductor $384$
Sign $-0.816 - 0.577i$
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  + (−4.24 − 3i)3-s + 5.65·5-s + 16.9i·7-s + (8.99 + 25.4i)9-s + 30i·11-s − 72i·13-s + (−24 − 16.9i)15-s − 50.9i·17-s − 25.4·19-s + (50.9 − 71.9i)21-s − 144·23-s − 93·25-s + (38.1 − 134. i)27-s + 5.65·29-s + 220. i·31-s + ⋯
L(s)  = 1  + (−0.816 − 0.577i)3-s + 0.505·5-s + 0.916i·7-s + (0.333 + 0.942i)9-s + 0.822i·11-s − 1.53i·13-s + (−0.413 − 0.292i)15-s − 0.726i·17-s − 0.307·19-s + (0.529 − 0.748i)21-s − 1.30·23-s − 0.743·25-s + (0.272 − 0.962i)27-s + 0.0362·29-s + 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.816 - 0.577i$
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: \((2,\ 384,\ (\ :3/2),\ -0.816 - 0.577i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3078461414\)
\(L(\frac12)\) \(\approx\) \(0.3078461414\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.24 + 3i)T \)
good5 \( 1 - 5.65T + 125T^{2} \)
7 \( 1 - 16.9iT - 343T^{2} \)
11 \( 1 - 30iT - 1.33e3T^{2} \)
13 \( 1 + 72iT - 2.19e3T^{2} \)
17 \( 1 + 50.9iT - 4.91e3T^{2} \)
19 \( 1 + 25.4T + 6.85e3T^{2} \)
23 \( 1 + 144T + 1.21e4T^{2} \)
29 \( 1 - 5.65T + 2.43e4T^{2} \)
31 \( 1 - 220. iT - 2.97e4T^{2} \)
37 \( 1 - 72iT - 5.06e4T^{2} \)
41 \( 1 - 305. iT - 6.89e4T^{2} \)
43 \( 1 - 229.T + 7.95e4T^{2} \)
47 \( 1 + 576T + 1.03e5T^{2} \)
53 \( 1 + 514.T + 1.48e5T^{2} \)
59 \( 1 + 414iT - 2.05e5T^{2} \)
61 \( 1 - 504iT - 2.26e5T^{2} \)
67 \( 1 + 789.T + 3.00e5T^{2} \)
71 \( 1 + 720T + 3.57e5T^{2} \)
73 \( 1 + 178T + 3.89e5T^{2} \)
79 \( 1 - 967. iT - 4.93e5T^{2} \)
83 \( 1 - 438iT - 5.71e5T^{2} \)
89 \( 1 + 865. iT - 7.04e5T^{2} \)
97 \( 1 - 650T + 9.12e5T^{2} \)
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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

−11.42781723261180344783342380564, −10.31599360061126176686639583731, −9.702187823605219810596129867316, −8.343218639126631751172597323835, −7.51505926995414771743941616050, −6.32584564579157437845224434222, −5.61602339527864029087998131985, −4.74086984240893309868163592310, −2.78973873002070409550126386262, −1.61469915683566450054172956076, 0.11271055687836309254715393360, 1.73892460011998840207050988012, 3.77767869185827828662869807685, 4.43271045994656069647611883641, 5.89012125519471838152103540997, 6.39626650088952568682881263892, 7.63657467797307740312754406415, 8.955929288825778851079014502552, 9.799179679948583256411289085764, 10.56484910473972042310922415833

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