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Properties

Label 2-567-63.41-c1-0-13
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: $\chi_{567} (377, \cdot )$
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

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

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