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Properties

Label 1-1083-1083.407-r0-0-0
Degree $1$
Conductor $1083$
Sign $-0.351 + 0.936i$
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  + (0.904 − 0.426i)2-s + (0.635 − 0.771i)4-s + (0.191 + 0.981i)5-s + (−0.986 + 0.164i)7-s + (0.245 − 0.969i)8-s + (0.592 + 0.805i)10-s + (−0.546 + 0.837i)11-s + (−0.851 + 0.523i)13-s + (−0.821 + 0.569i)14-s + (−0.191 − 0.981i)16-s + (−0.635 − 0.771i)17-s + (0.879 + 0.475i)20-s + (−0.137 + 0.990i)22-s + (−0.0275 + 0.999i)23-s + (−0.926 + 0.376i)25-s + (−0.546 + 0.837i)26-s + ⋯
L(s)  = 1  + (0.904 − 0.426i)2-s + (0.635 − 0.771i)4-s + (0.191 + 0.981i)5-s + (−0.986 + 0.164i)7-s + (0.245 − 0.969i)8-s + (0.592 + 0.805i)10-s + (−0.546 + 0.837i)11-s + (−0.851 + 0.523i)13-s + (−0.821 + 0.569i)14-s + (−0.191 − 0.981i)16-s + (−0.635 − 0.771i)17-s + (0.879 + 0.475i)20-s + (−0.137 + 0.990i)22-s + (−0.0275 + 0.999i)23-s + (−0.926 + 0.376i)25-s + (−0.546 + 0.837i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1083\)    =    \(3 \cdot 19^{2}\)
Sign: $-0.351 + 0.936i$
Analytic conductor: \(5.02943\)
Root analytic conductor: \(5.02943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1083} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1083,\ (0:\ ),\ -0.351 + 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6333773078 + 0.9148736481i\)
\(L(\frac12)\) \(\approx\) \(0.6333773078 + 0.9148736481i\)
\(L(1)\) \(\approx\) \(1.263419620 + 0.07147093090i\)
\(L(1)\) \(\approx\) \(1.263419620 + 0.07147093090i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.904 - 0.426i)T \)
5 \( 1 + (0.191 + 0.981i)T \)
7 \( 1 + (-0.986 + 0.164i)T \)
11 \( 1 + (-0.546 + 0.837i)T \)
13 \( 1 + (-0.851 + 0.523i)T \)
17 \( 1 + (-0.635 - 0.771i)T \)
23 \( 1 + (-0.0275 + 0.999i)T \)
29 \( 1 + (0.350 - 0.936i)T \)
31 \( 1 + (0.0825 + 0.996i)T \)
37 \( 1 + (-0.546 + 0.837i)T \)
41 \( 1 + (-0.298 + 0.954i)T \)
43 \( 1 + (-0.592 + 0.805i)T \)
47 \( 1 + (-0.451 - 0.892i)T \)
53 \( 1 + (0.451 + 0.892i)T \)
59 \( 1 + (-0.298 + 0.954i)T \)
61 \( 1 + (-0.962 - 0.272i)T \)
67 \( 1 + (-0.716 - 0.697i)T \)
71 \( 1 + (-0.962 + 0.272i)T \)
73 \( 1 + (0.635 + 0.771i)T \)
79 \( 1 + (0.592 - 0.805i)T \)
83 \( 1 + (-0.945 + 0.324i)T \)
89 \( 1 + (0.635 - 0.771i)T \)
97 \( 1 + (-0.716 + 0.697i)T \)
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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

−21.32744575452515787513035506595, −20.53981061425866412875195658507, −19.8645596192282319977481769373, −19.13966029937610041347133073187, −17.823564621389807722351793836, −16.988316058272863268617914434341, −16.44105088652361462086616656303, −15.79570878363499474630338551759, −15.027115621080371250955907995406, −13.96859199834598348624002618213, −13.2685636681440746949846036397, −12.66748914351775615348070774834, −12.18648783199868449636764544615, −10.920207601953613303375013871713, −10.14545968253972591295829994391, −8.93650797989140050332699856532, −8.29530717071605903867438118681, −7.313110050880732652358820404430, −6.339159131543641158994166593936, −5.63921981117188731642756272310, −4.84429466404059342043055030942, −3.91919407041431139787266018292, −2.99893347914679099184926668642, −2.03038357291963884745500756863, −0.29349765299643250131949388983, 1.76409620408636166737897993787, 2.67847679492051825970220507306, 3.18308757202961721771180249414, 4.377573151027144713818574385701, 5.19527294923776070793076664457, 6.28289828231217933819130242960, 6.87585812920920071183540636478, 7.550953358699934788479770582652, 9.31318870979165354609260165544, 9.93537472726406343037527666786, 10.49743733640788492031150583699, 11.667965342372851911040523554398, 12.07960590007204156917987959925, 13.24721916048920533802764962397, 13.62673728102388029166377570360, 14.62039605452130698071823365203, 15.34664010156555853828853570, 15.828181264972107510258926500407, 16.95538382627826312888811902595, 18.06632898691293573893951306170, 18.70234584389400914351498366479, 19.65442719009406631860346872049, 19.9393466189038764914379283468, 21.25481992055403033987880542531, 21.67418963456482804906980771612

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