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Properties

Label 2-567-63.20-c1-0-15
Degree $2$
Conductor $567$
Sign $0.630 + 0.775i$
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 − 1.73i)4-s + (2 + 1.73i)7-s + (4.5 − 2.59i)13-s + (−1.99 + 3.46i)16-s − 5.19i·19-s + (2.5 − 4.33i)25-s + (0.999 − 5.19i)28-s + (9 − 5.19i)31-s − 11·37-s + (4 − 6.92i)43-s + (1.00 + 6.92i)49-s + (−9 − 5.19i)52-s + (13.5 + 7.79i)61-s + 7.99·64-s + (−2.5 − 4.33i)67-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)4-s + (0.755 + 0.654i)7-s + (1.24 − 0.720i)13-s + (−0.499 + 0.866i)16-s − 1.19i·19-s + (0.5 − 0.866i)25-s + (0.188 − 0.981i)28-s + (1.61 − 0.933i)31-s − 1.80·37-s + (0.609 − 1.05i)43-s + (0.142 + 0.989i)49-s + (−1.24 − 0.720i)52-s + (1.72 + 0.997i)61-s + 0.999·64-s + (−0.305 − 0.529i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.630 + 0.775i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.630 + 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29072 - 0.614171i\)
\(L(\frac12)\) \(\approx\) \(1.29072 - 0.614171i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 1.73i)T \)
good2 \( 1 + (1 + 1.73i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.5 + 2.59i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-9 + 5.19i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 11T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-13.5 - 7.79i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 15.5iT - 73T^{2} \)
79 \( 1 + (8.5 - 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (4.5 + 2.59i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.66174453442076490107849856037, −9.792635039232500839134158563523, −8.631174265785056806418123754405, −8.421112205860520235729250153707, −6.86539205183067490788031982889, −5.85306585121396278632541510528, −5.13949734779561967109687541268, −4.12369611000396327449666375077, −2.49772205195111361636446926241, −0.984617890730932138821312825415, 1.46878458416884065570138314693, 3.31771300176373423347907024641, 4.13917340566943113438696120691, 5.08791430922429896050538253908, 6.47014582724734271648677274731, 7.43428625947642816894114276850, 8.320186933738542966062270145530, 8.839544318817663542280157459757, 10.02650695254519338829399270359, 10.94568028476647276280431199318

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