Properties

Label 2-567-63.16-c1-0-17
Degree $2$
Conductor $567$
Sign $0.902 + 0.430i$
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 − 1.73i)4-s + (2.5 + 0.866i)7-s + (3.5 + 6.06i)13-s + (−1.99 − 3.46i)16-s + (3.5 − 6.06i)19-s − 5·25-s + (4 − 3.46i)28-s + (3.5 − 6.06i)31-s + (0.5 − 0.866i)37-s + (−2.5 + 4.33i)43-s + (5.5 + 4.33i)49-s + 14·52-s + (−7 − 12.1i)61-s − 7.99·64-s + (−5.5 + 9.52i)67-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s + (0.944 + 0.327i)7-s + (0.970 + 1.68i)13-s + (−0.499 − 0.866i)16-s + (0.802 − 1.39i)19-s − 25-s + (0.755 − 0.654i)28-s + (0.628 − 1.08i)31-s + (0.0821 − 0.142i)37-s + (−0.381 + 0.660i)43-s + (0.785 + 0.618i)49-s + 1.94·52-s + (−0.896 − 1.55i)61-s − 0.999·64-s + (−0.671 + 1.16i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.902 + 0.430i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.902 + 0.430i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80159 - 0.407253i\)
\(L(\frac12)\) \(\approx\) \(1.80159 - 0.407253i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-3.5 - 6.06i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7 - 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.00618687603374488100522142787, −9.681627654740260582367234391236, −9.118792271025037367087670882804, −8.017022069291099811284592791469, −6.94145440879291088465465266815, −6.14595830660429113042814811759, −5.14099481787855597373490971154, −4.19359300362076988884737933878, −2.42952941470636248367231201601, −1.35894369580060168673200234568, 1.50383301377128624658897602765, 3.08517246852557650342965615941, 3.93017140693550642543459058897, 5.29940359658242321547915691426, 6.24926640933444087429668999049, 7.60965164553810521912058029929, 7.938533874516159532729217850229, 8.754741901892339896806207403149, 10.24505384462205628742717488056, 10.75743606721848065063443148425

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