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Properties

Label 2-567-189.4-c1-0-13
Degree $2$
Conductor $567$
Sign $0.875 + 0.482i$
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  + (0.467 + 2.65i)2-s + (−4.92 + 1.79i)4-s + (−2.75 + 1.00i)5-s + (−0.609 − 2.57i)7-s + (−4.36 − 7.55i)8-s + (−3.94 − 6.82i)10-s + (2.20 + 0.803i)11-s + (−1.68 + 0.612i)13-s + (6.53 − 2.81i)14-s + (9.95 − 8.35i)16-s + (0.185 + 0.320i)17-s + (−0.496 + 0.860i)19-s + (11.7 − 9.87i)20-s + (−1.09 + 6.22i)22-s + (1.14 − 6.50i)23-s + ⋯
L(s)  = 1  + (0.330 + 1.87i)2-s + (−2.46 + 0.896i)4-s + (−1.23 + 0.448i)5-s + (−0.230 − 0.973i)7-s + (−1.54 − 2.67i)8-s + (−1.24 − 2.15i)10-s + (0.665 + 0.242i)11-s + (−0.467 + 0.170i)13-s + (1.74 − 0.753i)14-s + (2.48 − 2.08i)16-s + (0.0449 + 0.0777i)17-s + (−0.113 + 0.197i)19-s + (2.63 − 2.20i)20-s + (−0.233 + 1.32i)22-s + (0.239 − 1.35i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.875 + 0.482i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.875 + 0.482i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.106727 - 0.0274717i\)
\(L(\frac12)\) \(\approx\) \(0.106727 - 0.0274717i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.609 + 2.57i)T \)
good2 \( 1 + (-0.467 - 2.65i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (2.75 - 1.00i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (-2.20 - 0.803i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (1.68 - 0.612i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.185 - 0.320i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.496 - 0.860i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.14 + 6.50i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (6.76 + 2.46i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (3.45 - 1.25i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + 1.65T + 37T^{2} \)
41 \( 1 + (8.90 - 3.23i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.36 + 7.73i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-0.234 - 0.0853i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-4.11 + 7.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.89 + 3.26i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.813 - 0.296i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.371 + 2.10i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (0.743 - 1.28i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 9.96T + 73T^{2} \)
79 \( 1 + (-1.60 - 9.09i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-6.30 - 2.29i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (6.39 - 11.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.55 - 14.4i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.52846216957918010148732897706, −9.471661703519834380376909266011, −8.467826509750829744649720696368, −7.70154696366143614207489320325, −6.99990259609734551676390125459, −6.54779885439315341879653121864, −5.14246791622252235371427744575, −4.12275258225722509390004700139, −3.61922314822492914053353051460, −0.05834927370905931403424671107, 1.64193206022391490623218564270, 3.07891903814526374539068415673, 3.80360045995253095194342350329, 4.83406367972990661515626062121, 5.72609885530557703275015268497, 7.51856308225686732324309106992, 8.727833719276162354419897976509, 9.150549761496522549352530759555, 10.07671670769645032855496188282, 11.30313697679603994947559166251

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