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Properties

Label 2-1067-1067.648-c0-0-0
Degree $2$
Conductor $1067$
Sign $0.179 + 0.983i$
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  + (0.252 − 1.91i)3-s + (−0.258 + 0.965i)4-s + (1.88 − 0.123i)5-s + (−2.63 − 0.707i)9-s + (−0.793 − 0.608i)11-s + (1.78 + 0.739i)12-s + (0.239 − 3.65i)15-s + (−0.866 − 0.499i)16-s + (−0.369 + 1.85i)20-s + (0.0420 + 0.123i)23-s + (2.56 − 0.337i)25-s + (−1.28 + 3.09i)27-s + (0.758 + 0.0999i)31-s + (−1.36 + 1.36i)33-s + (1.36 − 2.36i)36-s + (1.69 + 0.576i)37-s + ⋯
L(s)  = 1  + (0.252 − 1.91i)3-s + (−0.258 + 0.965i)4-s + (1.88 − 0.123i)5-s + (−2.63 − 0.707i)9-s + (−0.793 − 0.608i)11-s + (1.78 + 0.739i)12-s + (0.239 − 3.65i)15-s + (−0.866 − 0.499i)16-s + (−0.369 + 1.85i)20-s + (0.0420 + 0.123i)23-s + (2.56 − 0.337i)25-s + (−1.28 + 3.09i)27-s + (0.758 + 0.0999i)31-s + (−1.36 + 1.36i)33-s + (1.36 − 2.36i)36-s + (1.69 + 0.576i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1067\)    =    \(11 \cdot 97\)
Sign: $0.179 + 0.983i$
Analytic conductor: \(0.532502\)
Root analytic conductor: \(0.729727\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1067} (648, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1067,\ (\ :0),\ 0.179 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.297346761\)
\(L(\frac12)\) \(\approx\) \(1.297346761\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.793 + 0.608i)T \)
97 \( 1 + (-0.608 - 0.793i)T \)
good2 \( 1 + (0.258 - 0.965i)T^{2} \)
3 \( 1 + (-0.252 + 1.91i)T + (-0.965 - 0.258i)T^{2} \)
5 \( 1 + (-1.88 + 0.123i)T + (0.991 - 0.130i)T^{2} \)
7 \( 1 + (0.608 - 0.793i)T^{2} \)
13 \( 1 + (0.991 - 0.130i)T^{2} \)
17 \( 1 + (-0.608 - 0.793i)T^{2} \)
19 \( 1 + (0.382 - 0.923i)T^{2} \)
23 \( 1 + (-0.0420 - 0.123i)T + (-0.793 + 0.608i)T^{2} \)
29 \( 1 + (0.130 + 0.991i)T^{2} \)
31 \( 1 + (-0.758 - 0.0999i)T + (0.965 + 0.258i)T^{2} \)
37 \( 1 + (-1.69 - 0.576i)T + (0.793 + 0.608i)T^{2} \)
41 \( 1 + (-0.130 - 0.991i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
53 \( 1 + (1.57 - 1.20i)T + (0.258 - 0.965i)T^{2} \)
59 \( 1 + (0.206 - 0.608i)T + (-0.793 - 0.608i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.25 - 0.835i)T + (0.382 - 0.923i)T^{2} \)
71 \( 1 + (-0.665 + 0.583i)T + (0.130 - 0.991i)T^{2} \)
73 \( 1 + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (0.707 - 0.707i)T^{2} \)
83 \( 1 + (0.608 + 0.793i)T^{2} \)
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

−9.618556279506474819361285219526, −8.883782561874200139924900067016, −8.157542628485092263410771135595, −7.50888579716390151974048904726, −6.45207250745491261810226285407, −6.03424066574292685402771366741, −4.99466386748032220841321957053, −2.96020472705014863822745960629, −2.51967165577021666062947054579, −1.33365117729265036454146320398, 2.05354353123625424157088121425, 2.97325375237041870821304015252, 4.53818808672788115874684150005, 5.02997496373146718150494271769, 5.76027586832824042518219162081, 6.42598134835164191721317069356, 8.204315494272098147247853319380, 9.193317003615901554952371105373, 9.726255673588281451247183271299, 9.956241109877349568370604301272

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