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Properties

Label 2-567-7.2-c1-0-14
Degree $2$
Conductor $567$
Sign $0.925 + 0.377i$
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  + (0.635 − 1.10i)2-s + (0.193 + 0.334i)4-s + (−0.776 + 1.34i)5-s + (−1.48 − 2.18i)7-s + 3.03·8-s + (0.986 + 1.70i)10-s + (1.60 + 2.77i)11-s + 4.78·13-s + (−3.35 + 0.245i)14-s + (1.53 − 2.66i)16-s + (1.05 + 1.83i)17-s + (2.43 − 4.21i)19-s − 0.600·20-s + 4.07·22-s + (−1.85 + 3.21i)23-s + ⋯
L(s)  = 1  + (0.449 − 0.777i)2-s + (0.0966 + 0.167i)4-s + (−0.347 + 0.601i)5-s + (−0.561 − 0.827i)7-s + 1.07·8-s + (0.311 + 0.540i)10-s + (0.483 + 0.838i)11-s + 1.32·13-s + (−0.895 + 0.0655i)14-s + (0.384 − 0.666i)16-s + (0.256 + 0.444i)17-s + (0.557 − 0.966i)19-s − 0.134·20-s + 0.869·22-s + (−0.386 + 0.669i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.925 + 0.377i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.925 + 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95755 - 0.383741i\)
\(L(\frac12)\) \(\approx\) \(1.95755 - 0.383741i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.48 + 2.18i)T \)
good2 \( 1 + (-0.635 + 1.10i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.776 - 1.34i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.60 - 2.77i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.78T + 13T^{2} \)
17 \( 1 + (-1.05 - 1.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.43 + 4.21i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.85 - 3.21i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.37T + 29T^{2} \)
31 \( 1 + (2.75 + 4.76i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0932 + 0.161i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 - 4.86T + 43T^{2} \)
47 \( 1 + (0.885 - 1.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.834 + 1.44i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.91 + 5.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.43 - 5.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.11 + 10.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + (5.93 + 10.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.654 + 1.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.346T + 83T^{2} \)
89 \( 1 + (-8.70 + 15.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−10.78958789824989485583737429729, −10.19338736311867262044126075988, −9.118712013280243512347807772656, −7.80184305754911631656153885338, −7.12872122025009841186282994764, −6.28456788687308062718394777076, −4.64391823298403364483246958232, −3.72090169189119196445488702546, −3.08586475292292406127240128571, −1.47614002420573172837877414546, 1.27029612115338170958390381798, 3.18258797816224937803177385089, 4.35698592983613903476626537350, 5.52375886919741236716929790023, 6.10371160406115555260904367355, 6.92694109612625626463266819120, 8.326533606272062263839739701806, 8.650441539090874686138066183632, 9.914937135912519136317531298097, 10.79852873796803815743191098573

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