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Properties

Label 2-567-63.41-c1-0-18
Degree $2$
Conductor $567$
Sign $0.987 + 0.158i$
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  + (−1.93 + 1.11i)2-s + (1.5 − 2.59i)4-s + (1.93 − 3.35i)5-s + (2.5 + 0.866i)7-s + 2.23i·8-s + 8.66i·10-s + (−1.93 + 1.11i)11-s + (3 + 1.73i)13-s + (−5.80 + 1.11i)14-s + (0.499 + 0.866i)16-s − 5.19i·19-s + (−5.80 − 10.0i)20-s + (2.5 − 4.33i)22-s + (1.93 + 1.11i)23-s + (−5.00 − 8.66i)25-s − 7.74·26-s + ⋯
L(s)  = 1  + (−1.36 + 0.790i)2-s + (0.750 − 1.29i)4-s + (0.866 − 1.50i)5-s + (0.944 + 0.327i)7-s + 0.790i·8-s + 2.73i·10-s + (−0.583 + 0.337i)11-s + (0.832 + 0.480i)13-s + (−1.55 + 0.298i)14-s + (0.124 + 0.216i)16-s − 1.19i·19-s + (−1.29 − 2.25i)20-s + (0.533 − 0.923i)22-s + (0.403 + 0.233i)23-s + (−1.00 − 1.73i)25-s − 1.51·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.987 + 0.158i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.987 + 0.158i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.934263 - 0.0744027i\)
\(L(\frac12)\) \(\approx\) \(0.934263 - 0.0744027i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
good2 \( 1 + (1.93 - 1.11i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.93 + 3.35i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.93 - 1.11i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3 - 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 + (-1.93 - 1.11i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.87 + 2.23i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-1.93 + 3.35i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.87 - 6.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.94iT - 53T^{2} \)
59 \( 1 + (3.87 - 6.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.1iT - 71T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.87 - 6.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + (-12 + 6.92i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.40794507973768113315661261814, −9.390299192777872276559479540684, −8.914620056335308285980158389962, −8.332936302741483501713077236940, −7.44118954379238510665287134487, −6.25469127971860898288268069103, −5.36808412690314150499155533024, −4.52983620796335113849128062076, −2.04054001506005170146877753108, −0.975915184238140800645085870741, 1.40442597066638605488782204109, 2.50991023953333009337863163064, 3.47323369269430713544560509064, 5.38210960989430636847805161569, 6.44596277746386511962424158404, 7.56600871008272304298893486481, 8.181478586777342219156801517357, 9.171291429528916097084636610892, 10.30249129867471304674916618023, 10.54262149584423230733616478096

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