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Properties

Label 2-1067-1067.43-c0-0-0
Degree $2$
Conductor $1067$
Sign $0.700 - 0.713i$
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  + (−1.67 + 0.448i)3-s + (−0.866 + 0.5i)4-s + (1.57 + 0.207i)5-s + (1.73 − 1.00i)9-s + (0.258 − 0.965i)11-s + (1.22 − 1.22i)12-s + (−2.72 + 0.358i)15-s + (0.499 − 0.866i)16-s + (−1.46 + 0.607i)20-s + (1.57 + 1.20i)23-s + (1.46 + 0.392i)25-s + (−1.22 + 1.22i)27-s + (−1.36 + 0.366i)31-s + 1.73i·33-s + (−0.999 + 1.73i)36-s + (0.965 − 0.741i)37-s + ⋯
L(s)  = 1  + (−1.67 + 0.448i)3-s + (−0.866 + 0.5i)4-s + (1.57 + 0.207i)5-s + (1.73 − 1.00i)9-s + (0.258 − 0.965i)11-s + (1.22 − 1.22i)12-s + (−2.72 + 0.358i)15-s + (0.499 − 0.866i)16-s + (−1.46 + 0.607i)20-s + (1.57 + 1.20i)23-s + (1.46 + 0.392i)25-s + (−1.22 + 1.22i)27-s + (−1.36 + 0.366i)31-s + 1.73i·33-s + (−0.999 + 1.73i)36-s + (0.965 − 0.741i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1067\)    =    \(11 \cdot 97\)
Sign: $0.700 - 0.713i$
Analytic conductor: \(0.532502\)
Root analytic conductor: \(0.729727\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1067} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1067,\ (\ :0),\ 0.700 - 0.713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6770371190\)
\(L(\frac12)\) \(\approx\) \(0.6770371190\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.258 + 0.965i)T \)
97 \( 1 + (0.258 + 0.965i)T \)
good2 \( 1 + (0.866 - 0.5i)T^{2} \)
3 \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \)
5 \( 1 + (-1.57 - 0.207i)T + (0.965 + 0.258i)T^{2} \)
7 \( 1 + (0.258 - 0.965i)T^{2} \)
13 \( 1 + (-0.965 - 0.258i)T^{2} \)
17 \( 1 + (0.258 + 0.965i)T^{2} \)
19 \( 1 + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (-1.57 - 1.20i)T + (0.258 + 0.965i)T^{2} \)
29 \( 1 + (0.965 + 0.258i)T^{2} \)
31 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (-0.965 + 0.741i)T + (0.258 - 0.965i)T^{2} \)
41 \( 1 + (0.965 + 0.258i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - iT - T^{2} \)
53 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (-1.25 + 0.965i)T + (0.258 - 0.965i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.0999 - 0.241i)T + (-0.707 + 0.707i)T^{2} \)
71 \( 1 + (0.158 + 1.20i)T + (-0.965 + 0.258i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + (0.258 + 0.965i)T^{2} \)
89 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
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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

−10.26421295126342532953861983532, −9.281292487665985865532229811973, −9.134460924922548503955329704024, −7.51708666574510679184573908963, −6.52051968443637247209205686934, −5.62964546847733770793581698069, −5.38684799047214820359235561233, −4.31319410654968337338382375503, −3.11557077648563762988359078382, −1.19023044759393411468960197866, 1.03788317397373678165767898827, 2.07945405916573941284544838293, 4.31542130235695522309759522924, 5.25936280408458308704793950725, 5.45168866815159500485707829638, 6.53318931417617212565007497647, 6.96268666856928737548730170891, 8.543575190059339528932034753009, 9.494428717186682453642808602434, 10.02524434915583999298916446472

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