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Properties

Label 2-567-21.20-c3-0-26
Degree $2$
Conductor $567$
Sign $-0.986 - 0.164i$
Analytic cond. $33.4540$
Root an. cond. $5.78395$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.40i·2-s − 21.1·4-s − 6.87·5-s + (3.04 − 18.2i)7-s − 71.1i·8-s − 37.1i·10-s + 12.6i·11-s + 26.0i·13-s + (98.6 + 16.4i)14-s + 215.·16-s + 124.·17-s + 41.3i·19-s + 145.·20-s − 68.1·22-s − 112. i·23-s + ⋯
L(s)  = 1  + 1.90i·2-s − 2.64·4-s − 0.615·5-s + (0.164 − 0.986i)7-s − 3.14i·8-s − 1.17i·10-s + 0.345i·11-s + 0.555i·13-s + (1.88 + 0.314i)14-s + 3.36·16-s + 1.77·17-s + 0.498i·19-s + 1.62·20-s − 0.660·22-s − 1.02i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.986 - 0.164i$
Analytic conductor: \(33.4540\)
Root analytic conductor: \(5.78395\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (566, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :3/2),\ -0.986 - 0.164i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.189572809\)
\(L(\frac12)\) \(\approx\) \(1.189572809\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-3.04 + 18.2i)T \)
good2 \( 1 - 5.40iT - 8T^{2} \)
5 \( 1 + 6.87T + 125T^{2} \)
11 \( 1 - 12.6iT - 1.33e3T^{2} \)
13 \( 1 - 26.0iT - 2.19e3T^{2} \)
17 \( 1 - 124.T + 4.91e3T^{2} \)
19 \( 1 - 41.3iT - 6.85e3T^{2} \)
23 \( 1 + 112. iT - 1.21e4T^{2} \)
29 \( 1 + 132. iT - 2.43e4T^{2} \)
31 \( 1 - 179. iT - 2.97e4T^{2} \)
37 \( 1 - 148.T + 5.06e4T^{2} \)
41 \( 1 + 187.T + 6.89e4T^{2} \)
43 \( 1 + 198.T + 7.95e4T^{2} \)
47 \( 1 - 184.T + 1.03e5T^{2} \)
53 \( 1 - 359. iT - 1.48e5T^{2} \)
59 \( 1 - 364.T + 2.05e5T^{2} \)
61 \( 1 - 347. iT - 2.26e5T^{2} \)
67 \( 1 - 364.T + 3.00e5T^{2} \)
71 \( 1 - 565. iT - 3.57e5T^{2} \)
73 \( 1 - 737. iT - 3.89e5T^{2} \)
79 \( 1 - 903.T + 4.93e5T^{2} \)
83 \( 1 + 764.T + 5.71e5T^{2} \)
89 \( 1 - 395.T + 7.04e5T^{2} \)
97 \( 1 - 281. iT - 9.12e5T^{2} \)
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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

−10.30393637715607533729250077228, −9.726636774040449798815543068403, −8.512311877387704139285605245893, −7.85182726475626911809690344337, −7.25717039670191752826425990947, −6.42563009958069463198972625999, −5.37450316738520431628387881365, −4.38145247312613580147339763201, −3.68365953311577156445168801144, −0.896674444001856176092385076007, 0.51553226717604440141504750788, 1.80093060479099483576767029218, 3.05574568006124945533758901225, 3.65501992806233287068459123306, 5.00191497884938758904833300465, 5.71387184268629250448548324443, 7.76209172157228057877576823058, 8.405211514149319636129775805013, 9.407081102117035828260235099319, 9.993797139833838896680227762829

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