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Properties

Label 2-567-63.5-c1-0-2
Degree $2$
Conductor $567$
Sign $0.617 + 0.786i$
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  − 2.49i·2-s − 4.20·4-s + (0.617 + 1.07i)5-s + (−1.10 + 2.40i)7-s + 5.49i·8-s + (2.66 − 1.53i)10-s + (4.75 + 2.74i)11-s + (2.57 + 1.48i)13-s + (5.99 + 2.74i)14-s + 5.26·16-s + (2.15 + 3.73i)17-s + (−4.80 − 2.77i)19-s + (−2.59 − 4.49i)20-s + (6.83 − 11.8i)22-s + (−1.41 + 0.815i)23-s + ⋯
L(s)  = 1  − 1.76i·2-s − 2.10·4-s + (0.276 + 0.478i)5-s + (−0.416 + 0.909i)7-s + 1.94i·8-s + (0.843 − 0.486i)10-s + (1.43 + 0.827i)11-s + (0.712 + 0.411i)13-s + (1.60 + 0.733i)14-s + 1.31·16-s + (0.523 + 0.906i)17-s + (−1.10 − 0.636i)19-s + (−0.580 − 1.00i)20-s + (1.45 − 2.52i)22-s + (−0.294 + 0.170i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.617 + 0.786i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.617 + 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18631 - 0.577094i\)
\(L(\frac12)\) \(\approx\) \(1.18631 - 0.577094i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.10 - 2.40i)T \)
good2 \( 1 + 2.49iT - 2T^{2} \)
5 \( 1 + (-0.617 - 1.07i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.75 - 2.74i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.57 - 1.48i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.15 - 3.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.80 + 2.77i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.41 - 0.815i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.440 + 0.254i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.13iT - 31T^{2} \)
37 \( 1 + (-1.53 + 2.65i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.177 + 0.306i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.0318 - 0.0552i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.86T + 47T^{2} \)
53 \( 1 + (-3.57 + 2.06i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.07T + 59T^{2} \)
61 \( 1 - 6.89iT - 61T^{2} \)
67 \( 1 + 12.3T + 67T^{2} \)
71 \( 1 + 4.63iT - 71T^{2} \)
73 \( 1 + (6.09 - 3.51i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 0.331T + 79T^{2} \)
83 \( 1 + (-3.69 - 6.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.71 + 2.97i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.85 + 2.22i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.60984005621868053731151336501, −10.01542888228377035191668605139, −8.991135860803239468851241369376, −8.704629374003188493722769607775, −6.83685172074830917690038562534, −5.97114463678397173036247948326, −4.47270849316680525110841577958, −3.65458407610018862166111800542, −2.49608697570097362326998558976, −1.53425412391842197804484969004, 0.863321666939330703535475038254, 3.65195642321809674671178372382, 4.45641482020643900944376512117, 5.77938756929569377930827014181, 6.26736326612701754376343684906, 7.18916829065180531006914364570, 8.081911073350109347391718497709, 8.917544219888520719824959162579, 9.534587372247696464534587311229, 10.64897231102535691806361407090

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