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Properties

Label 2-567-63.34-c0-0-4
Degree $2$
Conductor $567$
Sign $-0.642 + 0.766i$
Analytic cond. $0.282969$
Root an. cond. $0.531949$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 1.5i)2-s + (−1 − 1.73i)4-s + (0.5 − 0.866i)7-s − 1.73·8-s + (−0.866 + 1.5i)11-s + (−0.866 − 1.5i)14-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)22-s + (−0.5 + 0.866i)25-s − 2·28-s + 37-s + (−0.5 + 0.866i)43-s + 3.46·44-s + (−0.499 − 0.866i)49-s + (0.866 + 1.5i)50-s + ⋯
L(s)  = 1  + (0.866 − 1.5i)2-s + (−1 − 1.73i)4-s + (0.5 − 0.866i)7-s − 1.73·8-s + (−0.866 + 1.5i)11-s + (−0.866 − 1.5i)14-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)22-s + (−0.5 + 0.866i)25-s − 2·28-s + 37-s + (−0.5 + 0.866i)43-s + 3.46·44-s + (−0.499 − 0.866i)49-s + (0.866 + 1.5i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(0.282969\)
Root analytic conductor: \(0.531949\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :0),\ -0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.310255648\)
\(L(\frac12)\) \(\approx\) \(1.310255648\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - 1.73T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + 1.73T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−10.72910215475527708553435199235, −10.11936423580543528909565189911, −9.437446582901955994128742406264, −7.932688347135148715707617051939, −7.07694377720816525232744420434, −5.51248795983523926276543548167, −4.66325149480541360091081066356, −3.96380536730328281717604132855, −2.65263834414705071376110007358, −1.54709738285794055070059203012, 2.71344633266395598793524361095, 3.98089733349999342402068739449, 5.17315514685199033767009988699, 5.72195726086316350282665801117, 6.50865661514965648426302015988, 7.75426214266634970591606308215, 8.289832288317075230742893891688, 9.006778346829113659695310477539, 10.45843479373789525368718989571, 11.50350205861024794968165646576

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