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Properties

Label 2-567-21.20-c3-0-16
Degree $2$
Conductor $567$
Sign $-0.174 + 0.984i$
Analytic cond. $33.4540$
Root an. cond. $5.78395$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.93i·2-s − 16.3·4-s − 15.3·5-s + (−18.2 − 3.22i)7-s + 41.0i·8-s + 75.5i·10-s + 42.6i·11-s + 28.0i·13-s + (−15.8 + 89.9i)14-s + 71.9·16-s − 82.0·17-s − 113. i·19-s + 250.·20-s + 210.·22-s − 29.1i·23-s + ⋯
L(s)  = 1  − 1.74i·2-s − 2.04·4-s − 1.36·5-s + (−0.984 − 0.174i)7-s + 1.81i·8-s + 2.38i·10-s + 1.16i·11-s + 0.598i·13-s + (−0.303 + 1.71i)14-s + 1.12·16-s − 1.17·17-s − 1.36i·19-s + 2.79·20-s + 2.03·22-s − 0.264i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.174 + 0.984i$
Analytic conductor: \(33.4540\)
Root analytic conductor: \(5.78395\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (566, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :3/2),\ -0.174 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5067112390\)
\(L(\frac12)\) \(\approx\) \(0.5067112390\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.2 + 3.22i)T \)
good2 \( 1 + 4.93iT - 8T^{2} \)
5 \( 1 + 15.3T + 125T^{2} \)
11 \( 1 - 42.6iT - 1.33e3T^{2} \)
13 \( 1 - 28.0iT - 2.19e3T^{2} \)
17 \( 1 + 82.0T + 4.91e3T^{2} \)
19 \( 1 + 113. iT - 6.85e3T^{2} \)
23 \( 1 + 29.1iT - 1.21e4T^{2} \)
29 \( 1 + 19.1iT - 2.43e4T^{2} \)
31 \( 1 + 112. iT - 2.97e4T^{2} \)
37 \( 1 - 69.2T + 5.06e4T^{2} \)
41 \( 1 + 484.T + 6.89e4T^{2} \)
43 \( 1 - 389.T + 7.95e4T^{2} \)
47 \( 1 + 106.T + 1.03e5T^{2} \)
53 \( 1 - 207. iT - 1.48e5T^{2} \)
59 \( 1 + 195.T + 2.05e5T^{2} \)
61 \( 1 - 515. iT - 2.26e5T^{2} \)
67 \( 1 + 442.T + 3.00e5T^{2} \)
71 \( 1 + 640. iT - 3.57e5T^{2} \)
73 \( 1 + 528. iT - 3.89e5T^{2} \)
79 \( 1 + 41.4T + 4.93e5T^{2} \)
83 \( 1 - 1.10e3T + 5.71e5T^{2} \)
89 \( 1 + 121.T + 7.04e5T^{2} \)
97 \( 1 + 864. iT - 9.12e5T^{2} \)
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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

−10.31512709587250452989258700813, −9.357558163471270843307181939873, −8.834335567876710709605876792250, −7.47853116422723721510053818627, −6.64118410423106765728625298159, −4.61205150548528609551890004350, −4.23139434387842931600637345785, −3.16655132451277782840700046030, −2.15050360040607611462997447766, −0.47468464864035875713119356565, 0.35732146655712961137299880866, 3.29626119546285431607528134165, 4.07738678445861778518415731960, 5.33038838785219026458974315066, 6.21311618178411993722521957689, 6.94553396136819399620729769931, 7.927929979541873656542943557312, 8.419715492625980189444839513464, 9.232460996714485488554133101936, 10.43973726738905841166404592395

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