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Properties

Label 1-33e2-1089.857-r0-0-0
Degree $1$
Conductor $1089$
Sign $-0.945 - 0.325i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
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  + (0.580 − 0.814i)2-s + (−0.327 − 0.945i)4-s + (−0.235 + 0.971i)5-s + (−0.928 + 0.371i)7-s + (−0.959 − 0.281i)8-s + (0.654 + 0.755i)10-s + (0.327 + 0.945i)13-s + (−0.235 + 0.971i)14-s + (−0.786 + 0.618i)16-s + (0.841 − 0.540i)17-s + (−0.841 − 0.540i)19-s + (0.995 − 0.0950i)20-s + (−0.928 − 0.371i)23-s + (−0.888 − 0.458i)25-s + (0.959 + 0.281i)26-s + ⋯
L(s)  = 1  + (0.580 − 0.814i)2-s + (−0.327 − 0.945i)4-s + (−0.235 + 0.971i)5-s + (−0.928 + 0.371i)7-s + (−0.959 − 0.281i)8-s + (0.654 + 0.755i)10-s + (0.327 + 0.945i)13-s + (−0.235 + 0.971i)14-s + (−0.786 + 0.618i)16-s + (0.841 − 0.540i)17-s + (−0.841 − 0.540i)19-s + (0.995 − 0.0950i)20-s + (−0.928 − 0.371i)23-s + (−0.888 − 0.458i)25-s + (0.959 + 0.281i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.945 - 0.325i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.945 - 0.325i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1263934543 - 0.7549630120i\)
\(L(\frac12)\) \(\approx\) \(0.1263934543 - 0.7549630120i\)
\(L(1)\) \(\approx\) \(0.8818000482 - 0.4062870054i\)
\(L(1)\) \(\approx\) \(0.8818000482 - 0.4062870054i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.580 - 0.814i)T \)
5 \( 1 + (-0.235 + 0.971i)T \)
7 \( 1 + (-0.928 + 0.371i)T \)
13 \( 1 + (0.327 + 0.945i)T \)
17 \( 1 + (0.841 - 0.540i)T \)
19 \( 1 + (-0.841 - 0.540i)T \)
23 \( 1 + (-0.928 - 0.371i)T \)
29 \( 1 + (0.0475 - 0.998i)T \)
31 \( 1 + (0.981 - 0.189i)T \)
37 \( 1 + (-0.654 - 0.755i)T \)
41 \( 1 + (-0.995 - 0.0950i)T \)
43 \( 1 + (-0.235 - 0.971i)T \)
47 \( 1 + (0.995 - 0.0950i)T \)
53 \( 1 + (0.142 - 0.989i)T \)
59 \( 1 + (-0.580 - 0.814i)T \)
61 \( 1 + (0.995 - 0.0950i)T \)
67 \( 1 + (-0.995 - 0.0950i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (0.142 + 0.989i)T \)
79 \( 1 + (-0.723 - 0.690i)T \)
83 \( 1 + (0.928 - 0.371i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (0.235 + 0.971i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.91916878200519441061298402473, −21.055724000197690606703870583266, −20.36329084578884483979321886392, −19.58556080396701193163444826481, −18.62948322865824279008471474766, −17.54650087454883507126117287254, −16.89406916320466441018460853689, −16.293127260474110635268871438654, −15.62799385624993163524913640360, −14.89821207764900102593942063894, −13.78741947572983147294602404453, −13.21787038396830123750273434781, −12.41898795043890727004559602951, −12.04669728804018593204794871056, −10.52689412345386568572898718740, −9.73047409414829787799697308084, −8.60680118721104248592619313427, −8.12553981791460048777311108375, −7.20825118791579855917291917393, −6.13038082992694790663167133000, −5.60992252790320122101371798100, −4.53382225351119358485089197377, −3.74333841544591459719109359320, −2.99413868394954257269703884855, −1.27267127801966313851158181894, 0.26503094263149747124417194648, 1.99753434910539153076552976194, 2.68456027920041254868624308176, 3.60477347030498151205735476701, 4.26844369205900594345111643046, 5.56054566045959780572906145403, 6.389594818607498903988078849404, 6.95217138682703889429150839038, 8.36978160134766191388337814690, 9.38616181197663472250625934702, 10.08950470995319592825567573165, 10.759521892354974483390502825461, 11.869466968957772611784676818797, 12.059682912217050750648087313236, 13.3273916668959693183765776011, 13.877935529203795530307983143754, 14.68130311460632627459325790636, 15.49782084384699908130136001740, 16.1240189631344952955353402963, 17.369531580797363434578466847188, 18.42681944308455246336255350607, 19.03123952412485433597155615526, 19.29055748402684707726385172593, 20.37585356138762161869595956499, 21.22442042830008816172108447118

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