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Properties

Label 1-1083-1083.905-r0-0-0
Degree $1$
Conductor $1083$
Sign $-0.669 - 0.742i$
Analytic cond. $5.02943$
Root an. cond. $5.02943$
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  + (0.451 + 0.892i)2-s + (−0.592 + 0.805i)4-s + (0.298 + 0.954i)5-s + (−0.401 + 0.915i)7-s + (−0.986 − 0.164i)8-s + (−0.716 + 0.697i)10-s + (−0.789 + 0.614i)11-s + (0.191 + 0.981i)13-s + (−0.998 + 0.0550i)14-s + (−0.298 − 0.954i)16-s + (0.592 + 0.805i)17-s + (−0.945 − 0.324i)20-s + (−0.904 − 0.426i)22-s + (0.754 + 0.656i)23-s + (−0.821 + 0.569i)25-s + (−0.789 + 0.614i)26-s + ⋯
L(s)  = 1  + (0.451 + 0.892i)2-s + (−0.592 + 0.805i)4-s + (0.298 + 0.954i)5-s + (−0.401 + 0.915i)7-s + (−0.986 − 0.164i)8-s + (−0.716 + 0.697i)10-s + (−0.789 + 0.614i)11-s + (0.191 + 0.981i)13-s + (−0.998 + 0.0550i)14-s + (−0.298 − 0.954i)16-s + (0.592 + 0.805i)17-s + (−0.945 − 0.324i)20-s + (−0.904 − 0.426i)22-s + (0.754 + 0.656i)23-s + (−0.821 + 0.569i)25-s + (−0.789 + 0.614i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1083\)    =    \(3 \cdot 19^{2}\)
Sign: $-0.669 - 0.742i$
Analytic conductor: \(5.02943\)
Root analytic conductor: \(5.02943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1083} (905, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1083,\ (0:\ ),\ -0.669 - 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5679108124 + 1.277043740i\)
\(L(\frac12)\) \(\approx\) \(-0.5679108124 + 1.277043740i\)
\(L(1)\) \(\approx\) \(0.6001166393 + 0.9760856416i\)
\(L(1)\) \(\approx\) \(0.6001166393 + 0.9760856416i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.451 + 0.892i)T \)
5 \( 1 + (0.298 + 0.954i)T \)
7 \( 1 + (-0.401 + 0.915i)T \)
11 \( 1 + (-0.789 + 0.614i)T \)
13 \( 1 + (0.191 + 0.981i)T \)
17 \( 1 + (0.592 + 0.805i)T \)
23 \( 1 + (0.754 + 0.656i)T \)
29 \( 1 + (0.993 - 0.110i)T \)
31 \( 1 + (-0.546 - 0.837i)T \)
37 \( 1 + (-0.789 + 0.614i)T \)
41 \( 1 + (0.0275 - 0.999i)T \)
43 \( 1 + (0.716 + 0.697i)T \)
47 \( 1 + (0.926 + 0.376i)T \)
53 \( 1 + (-0.926 - 0.376i)T \)
59 \( 1 + (0.0275 - 0.999i)T \)
61 \( 1 + (0.635 + 0.771i)T \)
67 \( 1 + (-0.350 - 0.936i)T \)
71 \( 1 + (0.635 - 0.771i)T \)
73 \( 1 + (-0.592 - 0.805i)T \)
79 \( 1 + (-0.716 - 0.697i)T \)
83 \( 1 + (0.677 + 0.735i)T \)
89 \( 1 + (-0.592 + 0.805i)T \)
97 \( 1 + (-0.350 + 0.936i)T \)
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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

−20.817277553981304231151365291735, −20.36738308773426890299354476870, −19.66981220277993469207441628145, −18.801414021488938467432868702717, −17.95130563273427201994586947212, −17.16083066728100392584705184065, −16.19142461771792532260445773717, −15.6024596209075157184566510903, −14.220660839106732683294971348942, −13.78400026807923139735245624942, −12.799135767366257069452598079262, −12.65304973271190239212897214404, −11.42508856445694120245835913435, −10.48057635784973486412334569876, −10.0998000152055128145802104198, −9.02402959881079339617237200740, −8.30655432314672601803095012515, −7.15130016435151662546709234597, −5.86539446191594910964711843722, −5.23116813068399588885916748216, −4.43517006609923899571874816715, −3.3576503124115482166125898256, −2.660941975708181937193663241606, −1.19527698400735847189201397064, −0.53911292000595964814430852466, 1.99213765318987243941936009590, 2.90899832834576122935530655254, 3.75196777221652608775719772028, 4.92108999787787370787354275915, 5.78511810375908127296637416850, 6.43837137626043012272085025203, 7.23482506820351940313684478342, 8.056663943601633004351435947439, 9.11001800972530664574679001066, 9.76492869625879072028195378491, 10.83451679548959067356990869161, 11.89888624535241915582343648531, 12.6084137979339332190925809376, 13.45676524938675413395404149277, 14.22849329854755654945842897576, 15.03172889486673119635330341228, 15.49683829890131499855371106658, 16.30215921744977072082047070441, 17.33666244841289788889289136348, 17.90379956625988189671445502795, 18.90721418188530069734583534253, 19.083478656637331402402860708634, 20.846212394114991523100993740270, 21.35475208808524234648288161009, 22.0718487185915282029411705132

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