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Properties

Label 2-864-27.7-c1-0-33
Degree $2$
Conductor $864$
Sign $-0.999 - 0.00893i$
Analytic cond. $6.89907$
Root an. cond. $2.62660$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.523 − 1.65i)3-s + (−0.649 − 3.68i)5-s + (1.60 + 1.34i)7-s + (−2.45 + 1.72i)9-s + (0.730 − 4.14i)11-s + (−2.93 − 1.06i)13-s + (−5.73 + 3.00i)15-s + (−0.103 − 0.179i)17-s + (1.10 − 1.90i)19-s + (1.38 − 3.35i)21-s + (6.25 − 5.24i)23-s + (−8.43 + 3.06i)25-s + (4.14 + 3.14i)27-s + (−8.80 + 3.20i)29-s + (−3.28 + 2.75i)31-s + ⋯
L(s)  = 1  + (−0.302 − 0.953i)3-s + (−0.290 − 1.64i)5-s + (0.606 + 0.509i)7-s + (−0.817 + 0.576i)9-s + (0.220 − 1.24i)11-s + (−0.815 − 0.296i)13-s + (−1.48 + 0.774i)15-s + (−0.0251 − 0.0435i)17-s + (0.252 − 0.437i)19-s + (0.301 − 0.732i)21-s + (1.30 − 1.09i)23-s + (−1.68 + 0.613i)25-s + (0.796 + 0.604i)27-s + (−1.63 + 0.595i)29-s + (−0.590 + 0.495i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.999 - 0.00893i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.999 - 0.00893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00451734 + 1.01115i\)
\(L(\frac12)\) \(\approx\) \(0.00451734 + 1.01115i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.523 + 1.65i)T \)
good5 \( 1 + (0.649 + 3.68i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-1.60 - 1.34i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.730 + 4.14i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (2.93 + 1.06i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.103 + 0.179i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.10 + 1.90i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.25 + 5.24i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (8.80 - 3.20i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (3.28 - 2.75i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-2.93 - 5.08i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.94 - 2.88i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.669 - 3.79i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (4.35 + 3.65i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 + (-1.72 - 9.76i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (6.65 + 5.58i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-1.02 - 0.373i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-2.33 - 4.04i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.90 + 8.49i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.47 - 0.900i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-6.67 + 2.42i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-5.28 + 9.15i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.431 - 2.44i)T + (-91.1 - 33.1i)T^{2} \)
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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

−9.359793998112971240408219244628, −8.740903454395560176293284908550, −8.154763761216746842040996765090, −7.34723030917577766197159541969, −6.12313985548148642697249660180, −5.22889049419249374044040807526, −4.74979476046174645446779360843, −3.04177112045749886929486432870, −1.59440679782965593355531692457, −0.51385120415915739733804357921, 2.17180291148133568265736354634, 3.43114743819134481663785511496, 4.20000779579036008477572538271, 5.19118383208083639379252330823, 6.30304616333920442083221700981, 7.41492643263217023741461455767, 7.58135988063886985959754188276, 9.474009952045537386826043255760, 9.623678981875283844247382203428, 10.83516954940849446005250662762

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