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Properties

Label 2-567-7.4-c1-0-16
Degree $2$
Conductor $567$
Sign $-0.739 + 0.673i$
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  + (−0.920 − 1.59i)2-s + (−0.695 + 1.20i)4-s + (0.667 + 1.15i)5-s + (0.640 − 2.56i)7-s − 1.12·8-s + (1.22 − 2.12i)10-s + (0.756 − 1.31i)11-s + 5.17·13-s + (−4.68 + 1.34i)14-s + (2.42 + 4.19i)16-s + (−0.774 + 1.34i)17-s + (−1.25 − 2.16i)19-s − 1.85·20-s − 2.78·22-s + (−3.68 − 6.37i)23-s + ⋯
L(s)  = 1  + (−0.650 − 1.12i)2-s + (−0.347 + 0.601i)4-s + (0.298 + 0.516i)5-s + (0.242 − 0.970i)7-s − 0.396·8-s + (0.388 − 0.673i)10-s + (0.228 − 0.395i)11-s + 1.43·13-s + (−1.25 + 0.358i)14-s + (0.605 + 1.04i)16-s + (−0.187 + 0.325i)17-s + (−0.287 − 0.497i)19-s − 0.414·20-s − 0.593·22-s + (−0.767 − 1.32i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.739 + 0.673i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.739 + 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.381681 - 0.985247i\)
\(L(\frac12)\) \(\approx\) \(0.381681 - 0.985247i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.640 + 2.56i)T \)
good2 \( 1 + (0.920 + 1.59i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.667 - 1.15i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.756 + 1.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.17T + 13T^{2} \)
17 \( 1 + (0.774 - 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.25 + 2.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.68 + 6.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.0619T + 29T^{2} \)
31 \( 1 + (-1.92 + 3.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.281 + 0.487i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.02T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 + (4.75 + 8.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.755 - 1.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.22 - 7.31i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.61 + 2.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.46 - 6.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (1.37 - 2.38i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.95 - 5.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.60T + 83T^{2} \)
89 \( 1 + (0.703 + 1.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−10.56908533257647694146056631924, −9.866976079949065295641214536420, −8.734302370484738169706626684074, −8.159154215123379711624294338812, −6.67187976345575033702370808955, −6.07754797388168961921619120821, −4.31250856735579159500092218028, −3.37310206428250416129392449125, −2.13027730405071359139973465030, −0.799637964423740662669410121731, 1.60979761625620545671991949265, 3.36027428857783327435270249649, 4.95041994404021379995241237248, 5.87657727208669199415859539780, 6.46867667426073302257719035148, 7.71611071125842433159753650041, 8.393998638588858087365439951864, 9.135659108556937555674123837947, 9.684191256123238687681210273371, 11.07353238104095853288373307188

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