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Properties

Label 2-567-189.101-c1-0-2
Degree $2$
Conductor $567$
Sign $0.144 - 0.989i$
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  + (1.07 − 1.28i)2-s + (−0.142 − 0.809i)4-s + (−2.87 + 2.40i)5-s + (−1.86 − 1.87i)7-s + (1.71 + 0.989i)8-s + 6.29i·10-s + (−3.07 + 3.66i)11-s + (−1.10 + 3.04i)13-s + (−4.42 + 0.369i)14-s + (4.66 − 1.69i)16-s − 3.86·17-s + 1.52i·19-s + (2.35 + 1.97i)20-s + (1.39 + 7.90i)22-s + (−0.124 + 0.342i)23-s + ⋯
L(s)  = 1  + (0.763 − 0.909i)2-s + (−0.0713 − 0.404i)4-s + (−1.28 + 1.07i)5-s + (−0.704 − 0.709i)7-s + (0.606 + 0.349i)8-s + 1.99i·10-s + (−0.926 + 1.10i)11-s + (−0.307 + 0.844i)13-s + (−1.18 + 0.0987i)14-s + (1.16 − 0.424i)16-s − 0.938·17-s + 0.349i·19-s + (0.527 + 0.442i)20-s + (0.297 + 1.68i)22-s + (−0.0259 + 0.0713i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.144 - 0.989i$
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: \((2,\ 567,\ (\ :1/2),\ 0.144 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.735195 + 0.635567i\)
\(L(\frac12)\) \(\approx\) \(0.735195 + 0.635567i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.86 + 1.87i)T \)
good2 \( 1 + (-1.07 + 1.28i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (2.87 - 2.40i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (3.07 - 3.66i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (1.10 - 3.04i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 3.86T + 17T^{2} \)
19 \( 1 - 1.52iT - 19T^{2} \)
23 \( 1 + (0.124 - 0.342i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-2.29 - 6.30i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (0.692 - 0.122i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-2.75 + 4.76i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.793 + 0.288i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.0136 + 0.0775i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.617 - 3.50i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (7.25 + 4.18i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.36 + 0.495i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-2.82 - 0.497i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-3.09 + 2.60i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-14.1 + 8.19i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (10.4 - 6.02i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.00 - 0.844i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-12.7 + 4.64i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 8.72T + 89T^{2} \)
97 \( 1 + (4.18 + 0.738i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.90044147772396214509420496449, −10.59220574376889444984879438359, −9.555511582790519616839750396875, −7.999493160796316549830143202984, −7.28523479185489039083543206467, −6.66220420411409797885278423006, −4.82283715756038791205720680911, −4.08305928538442363845586602311, −3.26465884991924489904273095678, −2.25441982905229755770692808489, 0.41442405360295471212940593587, 2.96641929751821270332784904749, 4.15783564122414425887994130452, 5.06718910550030061210316907902, 5.77698558221475375399543257980, 6.79831005984977339625521777072, 8.021790033171305647341828323439, 8.304468029503531300851012174587, 9.481601498044164997718303546376, 10.67083748271718615543723931312

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