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Properties

Label 2-864-27.4-c1-0-21
Degree $2$
Conductor $864$
Sign $0.999 - 0.00198i$
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  + (−0.527 + 1.64i)3-s + (−0.220 + 1.24i)5-s + (2.06 − 1.73i)7-s + (−2.44 − 1.74i)9-s + (−0.816 − 4.63i)11-s + (5.12 − 1.86i)13-s + (−1.94 − 1.02i)15-s + (2.58 − 4.48i)17-s + (1.60 + 2.78i)19-s + (1.77 + 4.32i)21-s + (−5.70 − 4.78i)23-s + (3.18 + 1.16i)25-s + (4.16 − 3.11i)27-s + (4.64 + 1.69i)29-s + (−5.45 − 4.57i)31-s + ⋯
L(s)  = 1  + (−0.304 + 0.952i)3-s + (−0.0984 + 0.558i)5-s + (0.781 − 0.655i)7-s + (−0.814 − 0.580i)9-s + (−0.246 − 1.39i)11-s + (1.42 − 0.516i)13-s + (−0.501 − 0.263i)15-s + (0.627 − 1.08i)17-s + (0.368 + 0.638i)19-s + (0.386 + 0.943i)21-s + (−1.18 − 0.998i)23-s + (0.637 + 0.232i)25-s + (0.800 − 0.598i)27-s + (0.862 + 0.314i)29-s + (−0.979 − 0.822i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.999 - 0.00198i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.999 - 0.00198i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48932 + 0.00147697i\)
\(L(\frac12)\) \(\approx\) \(1.48932 + 0.00147697i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.527 - 1.64i)T \)
good5 \( 1 + (0.220 - 1.24i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-2.06 + 1.73i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.816 + 4.63i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-5.12 + 1.86i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-2.58 + 4.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.60 - 2.78i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.70 + 4.78i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-4.64 - 1.69i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (5.45 + 4.57i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (4.05 - 7.03i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.89 + 2.14i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.35 - 7.68i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (3.37 - 2.83i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 5.40T + 53T^{2} \)
59 \( 1 + (-0.563 + 3.19i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-2.83 + 2.37i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-12.7 + 4.62i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-1.66 + 2.88i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.27 + 7.41i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.44 - 2.70i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (3.51 + 1.27i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-8.80 - 15.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.21 + 12.5i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.40306424288928676299795306188, −9.461760784361899545594605917776, −8.340024593468144017037874886703, −7.920904178456872311976111873443, −6.48050936295050272078729614212, −5.76137819303580262180970672827, −4.81376210491717275327564662189, −3.68749849121849003563218457507, −3.04761602004529276662942834246, −0.873563614441314345892072395922, 1.37712451051116648206484073834, 2.14086553789148331928915768989, 3.86812665222087942162372730250, 5.06804032899915283800943551553, 5.73596930946052147884666360733, 6.76486875814830892472314603175, 7.67500815171143417698258575336, 8.416746473589605479931813474470, 9.034934362742060209220486606708, 10.28252146768048484626518558360

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