Properties

Label 2-864-72.11-c1-0-6
Degree $2$
Conductor $864$
Sign $0.476 + 0.879i$
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  + (−1.60 − 2.78i)5-s + (1.82 + 1.05i)7-s + (3.47 + 2.00i)11-s + (0.341 − 0.197i)13-s − 1.20i·17-s + 1.62·19-s + (−2.74 − 4.75i)23-s + (−2.68 + 4.64i)25-s + (2.95 − 5.12i)29-s + (3.34 − 1.93i)31-s − 6.77i·35-s − 10.8i·37-s + (1.23 − 0.715i)41-s + (1.21 − 2.10i)43-s + (−0.792 + 1.37i)47-s + ⋯
L(s)  = 1  + (−0.719 − 1.24i)5-s + (0.688 + 0.397i)7-s + (1.04 + 0.605i)11-s + (0.0948 − 0.0547i)13-s − 0.292i·17-s + 0.372·19-s + (−0.572 − 0.990i)23-s + (−0.536 + 0.928i)25-s + (0.549 − 0.950i)29-s + (0.601 − 0.347i)31-s − 1.14i·35-s − 1.77i·37-s + (0.193 − 0.111i)41-s + (0.185 − 0.321i)43-s + (−0.115 + 0.200i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.476 + 0.879i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.476 + 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28944 - 0.768272i\)
\(L(\frac12)\) \(\approx\) \(1.28944 - 0.768272i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (1.60 + 2.78i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.82 - 1.05i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.47 - 2.00i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.341 + 0.197i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.20iT - 17T^{2} \)
19 \( 1 - 1.62T + 19T^{2} \)
23 \( 1 + (2.74 + 4.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.95 + 5.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.34 + 1.93i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 + (-1.23 + 0.715i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.21 + 2.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.792 - 1.37i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.07T + 53T^{2} \)
59 \( 1 + (2.29 - 1.32i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.18 - 4.72i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.60 - 4.51i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.69T + 71T^{2} \)
73 \( 1 - 9.49T + 73T^{2} \)
79 \( 1 + (-1.53 - 0.886i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.30 + 0.755i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.2iT - 89T^{2} \)
97 \( 1 + (-5.84 + 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.813428510273030435929277677247, −9.073510614178348788642819104692, −8.350032633891855486283862745025, −7.71374064090926675545779287406, −6.58010287367600598573722965124, −5.44923805784093786858081477430, −4.54015230602621336393050520765, −3.95834411340823807088668536044, −2.19859876916455307839529385027, −0.844513951492844499489900901994, 1.40075515597191603813504251654, 3.09156102043624856686917532823, 3.76917398519598632098849105085, 4.86307032821467869251892017887, 6.23545964607285063534694744038, 6.85360553383949491999231687585, 7.77763644534487773266503975419, 8.410614937207613078041702969666, 9.575838018419266545647065235883, 10.45133068243404949480774285310

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