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Properties

Label 2-384-12.11-c3-0-18
Degree $2$
Conductor $384$
Sign $0.381 - 0.924i$
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  + (4.80 + 1.98i)3-s − 11.9i·5-s + 22.6i·7-s + (19.1 + 19.0i)9-s − 61.9·11-s + 71.3·13-s + (23.7 − 57.5i)15-s + 74.3i·17-s + 108. i·19-s + (−44.9 + 109. i)21-s + 24.7·23-s − 18.6·25-s + (54.2 + 129. i)27-s − 84.1i·29-s + 130. i·31-s + ⋯
L(s)  = 1  + (0.924 + 0.381i)3-s − 1.07i·5-s + 1.22i·7-s + (0.709 + 0.705i)9-s − 1.69·11-s + 1.52·13-s + (0.408 − 0.990i)15-s + 1.06i·17-s + 1.30i·19-s + (−0.467 + 1.13i)21-s + 0.223·23-s − 0.149·25-s + (0.386 + 0.922i)27-s − 0.538i·29-s + 0.755i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.381 - 0.924i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 0.381 - 0.924i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.434152315\)
\(L(\frac12)\) \(\approx\) \(2.434152315\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.80 - 1.98i)T \)
good5 \( 1 + 11.9iT - 125T^{2} \)
7 \( 1 - 22.6iT - 343T^{2} \)
11 \( 1 + 61.9T + 1.33e3T^{2} \)
13 \( 1 - 71.3T + 2.19e3T^{2} \)
17 \( 1 - 74.3iT - 4.91e3T^{2} \)
19 \( 1 - 108. iT - 6.85e3T^{2} \)
23 \( 1 - 24.7T + 1.21e4T^{2} \)
29 \( 1 + 84.1iT - 2.43e4T^{2} \)
31 \( 1 - 130. iT - 2.97e4T^{2} \)
37 \( 1 - 397.T + 5.06e4T^{2} \)
41 \( 1 - 104. iT - 6.89e4T^{2} \)
43 \( 1 + 161. iT - 7.95e4T^{2} \)
47 \( 1 + 72.8T + 1.03e5T^{2} \)
53 \( 1 + 136. iT - 1.48e5T^{2} \)
59 \( 1 - 243.T + 2.05e5T^{2} \)
61 \( 1 + 358.T + 2.26e5T^{2} \)
67 \( 1 - 449. iT - 3.00e5T^{2} \)
71 \( 1 + 329.T + 3.57e5T^{2} \)
73 \( 1 + 925.T + 3.89e5T^{2} \)
79 \( 1 - 55.9iT - 4.93e5T^{2} \)
83 \( 1 - 928.T + 5.71e5T^{2} \)
89 \( 1 - 853. iT - 7.04e5T^{2} \)
97 \( 1 + 714.T + 9.12e5T^{2} \)
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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

−10.89579064514511732965998991252, −10.07008420146523242208573631421, −8.999422999678746202007739357622, −8.354934630334331435991235209018, −7.945543344722404555027525690550, −6.00223018880062197048097578854, −5.23833428007904049296482788935, −4.05253300630274218642535483797, −2.79448903512730261150043278178, −1.55009372673397896983237810779, 0.77017547506056586787946000948, 2.58095345392586171903929221782, 3.30592391466596761453424867991, 4.58519848198951454584868663823, 6.24819347621284965620683952891, 7.26239107370537701120941041233, 7.66313018214326511324468971898, 8.817261755564292880342019486336, 9.928586322936908052900126091133, 10.77550257547158700120036382364

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