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Properties

Label 2-567-63.4-c1-0-12
Degree $2$
Conductor $567$
Sign $0.160 - 0.987i$
Analytic cond. $4.52751$
Root an. cond. $2.12779$
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  + (−0.380 + 0.658i)2-s + (0.710 + 1.23i)4-s + 3.18·5-s + (1.85 + 1.88i)7-s − 2.60·8-s + (−1.21 + 2.09i)10-s + 2.23·11-s + (−1.85 + 3.20i)13-s + (−1.94 + 0.501i)14-s + (−0.430 + 0.746i)16-s + (2.80 − 4.85i)17-s + (−2.21 − 3.82i)19-s + (2.26 + 3.91i)20-s + (−0.851 + 1.47i)22-s − 0.942·23-s + ⋯
L(s)  = 1  + (−0.269 + 0.465i)2-s + (0.355 + 0.615i)4-s + 1.42·5-s + (0.699 + 0.714i)7-s − 0.920·8-s + (−0.382 + 0.663i)10-s + 0.675·11-s + (−0.513 + 0.889i)13-s + (−0.521 + 0.133i)14-s + (−0.107 + 0.186i)16-s + (0.679 − 1.17i)17-s + (−0.507 − 0.878i)19-s + (0.505 + 0.875i)20-s + (−0.181 + 0.314i)22-s − 0.196·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.160 - 0.987i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.160 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38787 + 1.18097i\)
\(L(\frac12)\) \(\approx\) \(1.38787 + 1.18097i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-1.85 - 1.88i)T \)
good2 \( 1 + (0.380 - 0.658i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 3.18T + 5T^{2} \)
11 \( 1 - 2.23T + 11T^{2} \)
13 \( 1 + (1.85 - 3.20i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.80 + 4.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.21 + 3.82i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.942T + 23T^{2} \)
29 \( 1 + (5.06 + 8.76i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.85 - 4.93i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.56 + 2.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.99 - 3.45i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.64 - 2.84i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.112 + 0.195i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.33 - 9.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.02 - 1.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.92 + 5.05i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.71 + 6.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.26T + 71T^{2} \)
73 \( 1 + (3.77 - 6.54i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.41 + 5.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.05 - 7.02i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.86 + 8.42i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.421 + 0.729i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.08154867837415797801678329971, −9.563284755807522872865413438575, −9.333829176005805999476262612995, −8.348929671055799080286239557817, −7.28526958398969761784159221961, −6.44492676370094300223276780371, −5.64985043215694704525581640847, −4.53770287125160215708016068487, −2.79085624074829133547276040809, −1.91423148426445952697347471268, 1.31306302187037574242251857078, 2.07433659389344923663832032849, 3.62235277381301415818517131117, 5.21456784379117538134893399473, 5.88133668714538253411718734391, 6.76924603004275759331464426834, 7.995105381602799497891182767144, 9.050966872764454357486253021594, 10.09881466620425055354516182593, 10.26652546680884787925417338330

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