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Properties

Label 2-567-189.101-c1-0-9
Degree $2$
Conductor $567$
Sign $0.415 - 0.909i$
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  + (−1.34 + 1.60i)2-s + (−0.412 − 2.34i)4-s + (−0.275 + 0.231i)5-s + (2.18 + 1.48i)7-s + (0.682 + 0.393i)8-s − 0.753i·10-s + (4.02 − 4.79i)11-s + (−0.429 + 1.18i)13-s + (−5.32 + 1.50i)14-s + (2.91 − 1.06i)16-s + 0.937·17-s − 7.64i·19-s + (0.655 + 0.550i)20-s + (2.27 + 12.8i)22-s + (0.584 − 1.60i)23-s + ⋯
L(s)  = 1  + (−0.950 + 1.13i)2-s + (−0.206 − 1.17i)4-s + (−0.123 + 0.103i)5-s + (0.826 + 0.562i)7-s + (0.241 + 0.139i)8-s − 0.238i·10-s + (1.21 − 1.44i)11-s + (−0.119 + 0.327i)13-s + (−1.42 + 0.402i)14-s + (0.729 − 0.265i)16-s + 0.227·17-s − 1.75i·19-s + (0.146 + 0.123i)20-s + (0.484 + 2.74i)22-s + (0.121 − 0.335i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.415 - 0.909i$
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: \((2,\ 567,\ (\ :1/2),\ 0.415 - 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.826242 + 0.530673i\)
\(L(\frac12)\) \(\approx\) \(0.826242 + 0.530673i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 - 1.48i)T \)
good2 \( 1 + (1.34 - 1.60i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (0.275 - 0.231i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-4.02 + 4.79i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.429 - 1.18i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 0.937T + 17T^{2} \)
19 \( 1 + 7.64iT - 19T^{2} \)
23 \( 1 + (-0.584 + 1.60i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.731 - 2.00i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-5.71 + 1.00i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-1.66 + 2.88i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.66 - 3.15i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.688 - 3.90i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.876 - 4.96i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-3.01 - 1.73i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.00 + 2.18i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (10.3 + 1.81i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (0.222 - 0.186i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-4.18 + 2.41i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.19 - 1.84i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.90 - 6.63i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (6.06 - 2.20i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 5.12T + 89T^{2} \)
97 \( 1 + (7.91 + 1.39i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.02562547684484915408140834689, −9.462617418304784446861261573998, −9.007878983388428407775612652493, −8.316204765285927293057564454220, −7.43099778456939876720984317647, −6.48597858592052970628599661544, −5.78603302966220326453492535545, −4.57882114578991403313352986798, −3.01022279044657108126929496435, −1.02644311633042057067477399283, 1.17642712091799465241832692403, 2.11127492439281890489690838543, 3.70694040923946490427619037062, 4.57554235240976542164558309805, 6.08095557434198311995283157010, 7.42727992388051691689357917423, 8.097634937319980900734723336002, 9.025773604730986229524922087629, 10.08068748391398543983573646504, 10.24313331545055526231441501749

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