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Properties

Label 2-567-9.2-c2-0-17
Degree $2$
Conductor $567$
Sign $-0.984 - 0.173i$
Analytic cond. $15.4496$
Root an. cond. $3.93060$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.40 + 1.96i)2-s + (5.71 + 9.89i)4-s + (−6.11 + 3.53i)5-s + (1.32 − 2.29i)7-s + 29.1i·8-s − 27.7·10-s + (4.30 + 2.48i)11-s + (−2.51 − 4.34i)13-s + (9.00 − 5.19i)14-s + (−34.4 + 59.6i)16-s + 5.51i·17-s − 18.2·19-s + (−69.9 − 40.3i)20-s + (9.75 + 16.9i)22-s + (9.25 − 5.34i)23-s + ⋯
L(s)  = 1  + (1.70 + 0.982i)2-s + (1.42 + 2.47i)4-s + (−1.22 + 0.706i)5-s + (0.188 − 0.327i)7-s + 3.64i·8-s − 2.77·10-s + (0.391 + 0.225i)11-s + (−0.193 − 0.334i)13-s + (0.642 − 0.371i)14-s + (−2.15 + 3.73i)16-s + 0.324i·17-s − 0.961·19-s + (−3.49 − 2.01i)20-s + (0.443 + 0.768i)22-s + (0.402 − 0.232i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.984 - 0.173i$
Analytic conductor: \(15.4496\)
Root analytic conductor: \(3.93060\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (512, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1),\ -0.984 - 0.173i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.649370144\)
\(L(\frac12)\) \(\approx\) \(3.649370144\)
\(L(2)\) 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 \( 1 + (-1.32 + 2.29i)T \)
good2 \( 1 + (-3.40 - 1.96i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (6.11 - 3.53i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-4.30 - 2.48i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (2.51 + 4.34i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 5.51iT - 289T^{2} \)
19 \( 1 + 18.2T + 361T^{2} \)
23 \( 1 + (-9.25 + 5.34i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-34.5 - 19.9i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-1.35 - 2.35i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 13.0T + 1.36e3T^{2} \)
41 \( 1 + (28.7 - 16.5i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-30.7 + 53.1i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-63.9 - 36.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 5.94iT - 2.80e3T^{2} \)
59 \( 1 + (-38.6 + 22.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-6.65 + 11.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-52.8 - 91.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 10.1iT - 5.04e3T^{2} \)
73 \( 1 - 133.T + 5.32e3T^{2} \)
79 \( 1 + (25.2 - 43.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-35.9 - 20.7i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 53.3iT - 7.92e3T^{2} \)
97 \( 1 + (48.9 - 84.7i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.20666244560358824632241662725, −10.61053191082472338789315672867, −8.588599762495130121053991383590, −7.907113338457617090633029965870, −7.01336269678441276398175140156, −6.59024496558964726539857076130, −5.30202278371160938426840852250, −4.28443231115829402301600671209, −3.69205218084266799345784885124, −2.61410307973257957390116774245, 0.798140574173173229208910323931, 2.27793775900261672546187433634, 3.53478196868033873532306849052, 4.35415104912226135124833022598, 4.98328426111141511096779944829, 6.11606022838916401683915217570, 7.09774141810241041784222727389, 8.414296105082407937639505145679, 9.489713452300937200127324308018, 10.62739134815245190681830077176

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