Properties

Label 2-864-24.11-c1-0-8
Degree $2$
Conductor $864$
Sign $0.892 - 0.451i$
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  + 3.42·5-s − 0.505i·7-s + 3.31i·11-s + 5.43i·13-s − 1.58i·17-s + 6.74·19-s − 4.30·23-s + 6.74·25-s − 2.55·29-s + 4.92i·31-s − 1.73i·35-s − 7.45i·37-s − 8.51i·41-s + 4·43-s − 5.10·47-s + ⋯
L(s)  = 1  + 1.53·5-s − 0.191i·7-s + 1.00i·11-s + 1.50i·13-s − 0.384i·17-s + 1.54·19-s − 0.897·23-s + 1.34·25-s − 0.473·29-s + 0.884i·31-s − 0.292i·35-s − 1.22i·37-s − 1.32i·41-s + 0.609·43-s − 0.744·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.892 - 0.451i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.892 - 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00111 + 0.476960i\)
\(L(\frac12)\) \(\approx\) \(2.00111 + 0.476960i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.42T + 5T^{2} \)
7 \( 1 + 0.505iT - 7T^{2} \)
11 \( 1 - 3.31iT - 11T^{2} \)
13 \( 1 - 5.43iT - 13T^{2} \)
17 \( 1 + 1.58iT - 17T^{2} \)
19 \( 1 - 6.74T + 19T^{2} \)
23 \( 1 + 4.30T + 23T^{2} \)
29 \( 1 + 2.55T + 29T^{2} \)
31 \( 1 - 4.92iT - 31T^{2} \)
37 \( 1 + 7.45iT - 37T^{2} \)
41 \( 1 + 8.51iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 5.10T + 47T^{2} \)
53 \( 1 + 0.875T + 53T^{2} \)
59 \( 1 + 6.63iT - 59T^{2} \)
61 \( 1 - 10.8iT - 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 9.40T + 71T^{2} \)
73 \( 1 + 3.74T + 73T^{2} \)
79 \( 1 + 6.44iT - 79T^{2} \)
83 \( 1 - 10.2iT - 83T^{2} \)
89 \( 1 + 3.75iT - 89T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−9.944331275496705268624811886623, −9.533642585026603059696052687436, −8.878224516542773732710747223726, −7.40974085387741903247650642554, −6.84979388065812458844720203631, −5.80835986202079524678315891563, −5.05250968492104765993016654074, −3.93412429774273470551505652582, −2.39032931511916959826561136391, −1.58058285366032570293682498013, 1.13770639634418469473904801268, 2.53210870945889866503666680983, 3.45022242313395660766719518460, 5.12071424278952179788837346623, 5.77285938986329475145816085425, 6.30192647023618178390957114805, 7.69966161101180956875952283829, 8.406319227887731630066465571644, 9.523463729033639652814765695432, 9.933809297147362298332415074661

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