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Properties

Label 2-567-189.101-c1-0-0
Degree $2$
Conductor $567$
Sign $0.377 + 0.925i$
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  + (−1.75 + 2.08i)2-s + (−0.940 − 5.33i)4-s + (−2.08 + 1.74i)5-s + (−0.122 + 2.64i)7-s + (8.05 + 4.65i)8-s − 7.40i·10-s + (−0.0911 + 0.108i)11-s + (−0.695 + 1.91i)13-s + (−5.29 − 4.88i)14-s + (−13.6 + 4.96i)16-s − 4.00·17-s + 3.04i·19-s + (11.2 + 9.47i)20-s + (−0.0670 − 0.380i)22-s + (0.449 − 1.23i)23-s + ⋯
L(s)  = 1  + (−1.23 + 1.47i)2-s + (−0.470 − 2.66i)4-s + (−0.931 + 0.781i)5-s + (−0.0463 + 0.998i)7-s + (2.84 + 1.64i)8-s − 2.34i·10-s + (−0.0274 + 0.0327i)11-s + (−0.192 + 0.530i)13-s + (−1.41 − 1.30i)14-s + (−3.40 + 1.24i)16-s − 0.970·17-s + 0.698i·19-s + (2.52 + 2.11i)20-s + (−0.0142 − 0.0810i)22-s + (0.0936 − 0.257i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.377 + 0.925i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.377 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0892191 - 0.0599668i\)
\(L(\frac12)\) \(\approx\) \(0.0892191 - 0.0599668i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.122 - 2.64i)T \)
good2 \( 1 + (1.75 - 2.08i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (2.08 - 1.74i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (0.0911 - 0.108i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.695 - 1.91i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 4.00T + 17T^{2} \)
19 \( 1 - 3.04iT - 19T^{2} \)
23 \( 1 + (-0.449 + 1.23i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (2.12 + 5.84i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (4.18 - 0.738i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-4.69 + 8.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.303 + 0.110i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.643 + 3.65i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.15 + 6.57i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-6.59 - 3.81i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.06 - 2.20i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (11.1 + 1.96i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (1.37 - 1.15i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (6.03 - 3.48i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.67 + 2.12i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (11.4 + 9.62i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-0.966 + 0.351i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 5.17T + 89T^{2} \)
97 \( 1 + (7.25 + 1.27i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.15042232218033412631045024794, −10.36238602859588296536387830451, −9.301579212611109357175167314630, −8.754795298948647257692457347929, −7.80188228549080934272581032195, −7.17798249169070824952853598656, −6.31079091789113851960367470807, −5.49084631023550192122111227386, −4.16065636769253926974219420638, −2.17930707537428840748659508052, 0.10105334363817669444513759141, 1.27629067560258458213442240908, 2.90805127001682678691111185565, 3.97174010120991882834799659003, 4.71613312086504508600646169651, 7.00036920054260489366172917668, 7.74944065035013051763593577045, 8.483984535996241089125654123381, 9.238820316038258853504510053987, 10.10046047965778791477394946756

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