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Properties

Label 2-3888-9.5-c0-0-1
Degree $2$
Conductor $3888$
Sign $0.766 - 0.642i$
Analytic cond. $1.94036$
Root an. cond. $1.39296$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)5-s + (−1.22 + 0.707i)11-s + (−0.5 + 0.866i)13-s − 1.41i·17-s + 19-s + (1.22 + 0.707i)23-s + (0.499 + 0.866i)25-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)43-s + (0.5 − 0.866i)49-s + 1.41i·53-s + 2·55-s + (1.22 + 0.707i)59-s + (−0.5 − 0.866i)61-s + (1.22 − 0.707i)65-s + ⋯
L(s)  = 1  + (−1.22 − 0.707i)5-s + (−1.22 + 0.707i)11-s + (−0.5 + 0.866i)13-s − 1.41i·17-s + 19-s + (1.22 + 0.707i)23-s + (0.499 + 0.866i)25-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)43-s + (0.5 − 0.866i)49-s + 1.41i·53-s + 2·55-s + (1.22 + 0.707i)59-s + (−0.5 − 0.866i)61-s + (1.22 − 0.707i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3888\)    =    \(2^{4} \cdot 3^{5}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(1.94036\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3888} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3888,\ (\ :0),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7835180262\)
\(L(\frac12)\) \(\approx\) \(0.7835180262\)
\(L(1)\) 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 \( 1 \)
good5 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.795276865014899770339536343004, −7.78525653846864962527410188400, −7.39477652206918054886690704579, −6.90229989814197865001515232120, −5.36358208762661813662214256012, −5.00054722187726560962981100692, −4.31760543643979077456664762134, −3.29653685652047644422763227432, −2.44619159362145186826583967988, −1.00003197452409065119794701104, 0.55354268833888863708761740332, 2.36038333770136979914717449043, 3.22581476028159468597205959802, 3.71373542318687590120962931788, 4.85179141651602118790912110506, 5.54818110689922294056004165096, 6.40804877195447876249405847604, 7.38651317820803082682951889885, 7.77036038026482133799481634066, 8.322591865529029905212000054052

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