Properties

Label 2-567-21.5-c1-0-4
Degree $2$
Conductor $567$
Sign $0.798 - 0.602i$
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  + (−1.88 − 1.08i)2-s + (1.35 + 2.35i)4-s + (−0.618 + 1.07i)5-s + (1.89 − 1.84i)7-s − 1.54i·8-s + (2.32 − 1.34i)10-s + (−0.968 + 0.558i)11-s + 5.39i·13-s + (−5.56 + 1.42i)14-s + (1.03 − 1.78i)16-s + (−0.913 − 1.58i)17-s + (−3.40 − 1.96i)19-s − 3.35·20-s + 2.42·22-s + (−0.167 − 0.0965i)23-s + ⋯
L(s)  = 1  + (−1.32 − 0.767i)2-s + (0.678 + 1.17i)4-s + (−0.276 + 0.479i)5-s + (0.715 − 0.698i)7-s − 0.547i·8-s + (0.735 − 0.424i)10-s + (−0.291 + 0.168i)11-s + 1.49i·13-s + (−1.48 + 0.379i)14-s + (0.257 − 0.446i)16-s + (−0.221 − 0.383i)17-s + (−0.782 − 0.451i)19-s − 0.750·20-s + 0.517·22-s + (−0.0348 − 0.0201i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.798 - 0.602i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.798 - 0.602i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.544021 + 0.182304i\)
\(L(\frac12)\) \(\approx\) \(0.544021 + 0.182304i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.89 + 1.84i)T \)
good2 \( 1 + (1.88 + 1.08i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.618 - 1.07i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.968 - 0.558i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.39iT - 13T^{2} \)
17 \( 1 + (0.913 + 1.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.40 + 1.96i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.167 + 0.0965i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.48iT - 29T^{2} \)
31 \( 1 + (4.27 - 2.47i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.81 - 10.0i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 - 3.42T + 43T^{2} \)
47 \( 1 + (-4.03 + 6.98i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.54 + 1.46i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.10 - 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.61 - 3.81i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.587 - 1.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.7iT - 71T^{2} \)
73 \( 1 + (-0.662 + 0.382i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.436 + 0.755i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 + (-0.705 + 1.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.92iT - 97T^{2} \)
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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

−10.81805631036792990377256869943, −10.11101075440411741048655955393, −9.011306069570793995427916098992, −8.547997373217025090168289832150, −7.25146138102722514026295687916, −6.98956724111203695429425633087, −5.09627440476407241072197881128, −3.93175205779639803142451956580, −2.52672522501435357428659871080, −1.37014424940157277594331273991, 0.54561953704939251768698563913, 2.23910648049427416815156878473, 4.09210301322524130831984330307, 5.50956184987141840270568189751, 6.11803995726198316126128082931, 7.57323256130789894260994348800, 8.055959201199986271427531789448, 8.674316567416125756623447811125, 9.500917277554281536596986816555, 10.54488650240929350130040503266

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