Properties

Label 2-1067-1067.483-c0-0-0
Degree $2$
Conductor $1067$
Sign $0.714 - 0.699i$
Analytic cond. $0.532502$
Root an. cond. $0.729727$
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  + (0.410 − 0.315i)3-s + (0.965 + 0.258i)4-s + (−1.18 + 0.583i)5-s + (−0.189 + 0.707i)9-s + (0.130 + 0.991i)11-s + (0.478 − 0.198i)12-s + (−0.301 + 0.612i)15-s + (0.866 + 0.499i)16-s + (−1.29 + 0.257i)20-s + (0.665 − 0.583i)23-s + (0.449 − 0.586i)25-s + (0.343 + 0.828i)27-s + (−0.465 − 0.607i)31-s + (0.366 + 0.366i)33-s + (−0.366 + 0.633i)36-s + (1.31 − 1.50i)37-s + ⋯
L(s)  = 1  + (0.410 − 0.315i)3-s + (0.965 + 0.258i)4-s + (−1.18 + 0.583i)5-s + (−0.189 + 0.707i)9-s + (0.130 + 0.991i)11-s + (0.478 − 0.198i)12-s + (−0.301 + 0.612i)15-s + (0.866 + 0.499i)16-s + (−1.29 + 0.257i)20-s + (0.665 − 0.583i)23-s + (0.449 − 0.586i)25-s + (0.343 + 0.828i)27-s + (−0.465 − 0.607i)31-s + (0.366 + 0.366i)33-s + (−0.366 + 0.633i)36-s + (1.31 − 1.50i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1067\)    =    \(11 \cdot 97\)
Sign: $0.714 - 0.699i$
Analytic conductor: \(0.532502\)
Root analytic conductor: \(0.729727\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1067} (483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1067,\ (\ :0),\ 0.714 - 0.699i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.186356176\)
\(L(\frac12)\) \(\approx\) \(1.186356176\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.130 - 0.991i)T \)
97 \( 1 + (-0.991 - 0.130i)T \)
good2 \( 1 + (-0.965 - 0.258i)T^{2} \)
3 \( 1 + (-0.410 + 0.315i)T + (0.258 - 0.965i)T^{2} \)
5 \( 1 + (1.18 - 0.583i)T + (0.608 - 0.793i)T^{2} \)
7 \( 1 + (0.991 - 0.130i)T^{2} \)
13 \( 1 + (0.608 - 0.793i)T^{2} \)
17 \( 1 + (-0.991 - 0.130i)T^{2} \)
19 \( 1 + (-0.382 - 0.923i)T^{2} \)
23 \( 1 + (-0.665 + 0.583i)T + (0.130 - 0.991i)T^{2} \)
29 \( 1 + (-0.793 - 0.608i)T^{2} \)
31 \( 1 + (0.465 + 0.607i)T + (-0.258 + 0.965i)T^{2} \)
37 \( 1 + (-1.31 + 1.50i)T + (-0.130 - 0.991i)T^{2} \)
41 \( 1 + (0.793 + 0.608i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
53 \( 1 + (-0.158 + 1.20i)T + (-0.965 - 0.258i)T^{2} \)
59 \( 1 + (1.13 + 0.991i)T + (0.130 + 0.991i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.357 + 0.534i)T + (-0.382 - 0.923i)T^{2} \)
71 \( 1 + (-0.0420 - 0.123i)T + (-0.793 + 0.608i)T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (0.707 + 0.707i)T^{2} \)
83 \( 1 + (0.991 + 0.130i)T^{2} \)
89 \( 1 + (0.707 + 0.707i)T^{2} \)
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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

−10.42422717828339876986867140323, −9.324129354930684522894584623172, −8.133346584732183973505856673327, −7.66522232873818150004097850544, −7.11951171448649727179978642667, −6.28025562132675441518871715236, −4.89279679117309501739128545588, −3.80064332523287782126324230810, −2.84739695623538615850853776995, −1.95563620545801859067888576266, 1.13133748402568996364604023283, 2.95347782456063275166243685701, 3.55649021007387176644791430962, 4.66025105523746843397473336212, 5.82001659626490439325347799950, 6.65151353011311174082775014160, 7.62655541195024249145186858367, 8.339260083780791468720131726850, 9.049509967897149613260872073652, 9.968870007791966219758939758351

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