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Properties

Label 2-3888-36.31-c0-0-5
Degree $2$
Conductor $3888$
Sign $0.642 + 0.766i$
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  + (1.5 − 0.866i)7-s + (1 − 1.73i)13-s + (0.5 + 0.866i)25-s + (−1.5 − 0.866i)31-s − 37-s + (−1.5 + 0.866i)43-s + (1 − 1.73i)49-s + (0.5 + 0.866i)61-s + (−1.5 − 0.866i)67-s + 73-s − 3.46i·91-s + (1 + 1.73i)97-s + (1.5 + 0.866i)103-s + 109-s + ⋯
L(s)  = 1  + (1.5 − 0.866i)7-s + (1 − 1.73i)13-s + (0.5 + 0.866i)25-s + (−1.5 − 0.866i)31-s − 37-s + (−1.5 + 0.866i)43-s + (1 − 1.73i)49-s + (0.5 + 0.866i)61-s + (−1.5 − 0.866i)67-s + 73-s − 3.46i·91-s + (1 + 1.73i)97-s + (1.5 + 0.866i)103-s + 109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3888\)    =    \(2^{4} \cdot 3^{5}\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(1.94036\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3888} (3727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3888,\ (\ :0),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.580458663\)
\(L(\frac12)\) \(\approx\) \(1.580458663\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.427173536858445908698911865016, −7.76684442644196227735990709751, −7.38107747237940171675678802994, −6.30420944179418688327969898291, −5.37715938237756924196454367049, −4.94544626569107273554113873968, −3.86810455554872039137197584415, −3.24880599759340016764385707698, −1.86147864468494075556742916431, −0.988164512003564007448958561414, 1.61343077709323413481975628117, 2.01231938625483712705564140244, 3.36921839121528907370734445260, 4.31883101176652438406258348613, 4.97844677055115349935867514711, 5.70169147781663000905255178721, 6.60131126873744608595057320890, 7.23949741095437286098092413238, 8.279573307832041009155757588814, 8.728803438317711565630801184048

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