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Properties

Label 2-384-24.11-c3-0-46
Degree $2$
Conductor $384$
Sign $-0.956 + 0.290i$
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  + (2.44 − 4.58i)3-s + 2.31·5-s − 29.6i·7-s + (−15 − 22.4i)9-s − 22.8i·11-s − 8.66i·13-s + (5.66 − 10.6i)15-s + 93.3i·17-s − 85.4·19-s + (−135. − 72.6i)21-s + 116.·23-s − 119.·25-s + (−139. + 13.7i)27-s + 103.·29-s + 10.6i·31-s + ⋯
L(s)  = 1  + (0.471 − 0.881i)3-s + 0.207·5-s − 1.60i·7-s + (−0.555 − 0.831i)9-s − 0.625i·11-s − 0.184i·13-s + (0.0975 − 0.182i)15-s + 1.33i·17-s − 1.03·19-s + (−1.41 − 0.755i)21-s + 1.05·23-s − 0.957·25-s + (−0.995 + 0.0979i)27-s + 0.662·29-s + 0.0614i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.956 + 0.290i$
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: \((2,\ 384,\ (\ :3/2),\ -0.956 + 0.290i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.675609944\)
\(L(\frac12)\) \(\approx\) \(1.675609944\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.44 + 4.58i)T \)
good5 \( 1 - 2.31T + 125T^{2} \)
7 \( 1 + 29.6iT - 343T^{2} \)
11 \( 1 + 22.8iT - 1.33e3T^{2} \)
13 \( 1 + 8.66iT - 2.19e3T^{2} \)
17 \( 1 - 93.3iT - 4.91e3T^{2} \)
19 \( 1 + 85.4T + 6.85e3T^{2} \)
23 \( 1 - 116.T + 1.21e4T^{2} \)
29 \( 1 - 103.T + 2.43e4T^{2} \)
31 \( 1 - 10.6iT - 2.97e4T^{2} \)
37 \( 1 + 380. iT - 5.06e4T^{2} \)
41 \( 1 - 257. iT - 6.89e4T^{2} \)
43 \( 1 - 359.T + 7.95e4T^{2} \)
47 \( 1 + 293.T + 1.03e5T^{2} \)
53 \( 1 + 679.T + 1.48e5T^{2} \)
59 \( 1 + 45.8iT - 2.05e5T^{2} \)
61 \( 1 + 595. iT - 2.26e5T^{2} \)
67 \( 1 + 206.T + 3.00e5T^{2} \)
71 \( 1 - 996.T + 3.57e5T^{2} \)
73 \( 1 + 593.T + 3.89e5T^{2} \)
79 \( 1 - 910. iT - 4.93e5T^{2} \)
83 \( 1 + 332. iT - 5.71e5T^{2} \)
89 \( 1 + 821. iT - 7.04e5T^{2} \)
97 \( 1 - 420.T + 9.12e5T^{2} \)
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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

−10.66110148309758461275744665984, −9.547562443661812915422149234832, −8.401575132594143408886565295737, −7.73776755511325035968566290425, −6.75068394759263364187687378896, −5.97861533587557226520053684740, −4.27224305671570967695660865122, −3.28179800491186981315394675913, −1.71950679098176083675573984777, −0.51619738184525490052247066138, 2.15768913745038328910112456723, 2.99849489829863590473139138468, 4.56163521634712613272520153953, 5.30674238011050316944826704178, 6.44417824976766295031178726038, 7.84451368165720201207785771503, 8.886364517185474937645477601543, 9.333510356771864339346113895158, 10.22783837758368908062288293187, 11.35187350214866567013092757365

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