Properties

Label 2-1067-1067.65-c0-0-0
Degree $2$
Conductor $1067$
Sign $-0.523 - 0.851i$
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  + (−1.91 + 0.252i)3-s + (0.258 + 0.965i)4-s + (−0.583 − 0.665i)5-s + (2.63 − 0.707i)9-s + (0.608 + 0.793i)11-s + (−0.739 − 1.78i)12-s + (1.28 + 1.12i)15-s + (−0.866 + 0.499i)16-s + (0.491 − 0.735i)20-s + (−1.34 + 0.665i)23-s + (0.0283 − 0.215i)25-s + (−3.09 + 1.28i)27-s + (0.241 + 1.83i)31-s + (−1.36 − 1.36i)33-s + (1.36 + 2.36i)36-s + (−0.837 + 1.69i)37-s + ⋯
L(s)  = 1  + (−1.91 + 0.252i)3-s + (0.258 + 0.965i)4-s + (−0.583 − 0.665i)5-s + (2.63 − 0.707i)9-s + (0.608 + 0.793i)11-s + (−0.739 − 1.78i)12-s + (1.28 + 1.12i)15-s + (−0.866 + 0.499i)16-s + (0.491 − 0.735i)20-s + (−1.34 + 0.665i)23-s + (0.0283 − 0.215i)25-s + (−3.09 + 1.28i)27-s + (0.241 + 1.83i)31-s + (−1.36 − 1.36i)33-s + (1.36 + 2.36i)36-s + (−0.837 + 1.69i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1067\)    =    \(11 \cdot 97\)
Sign: $-0.523 - 0.851i$
Analytic conductor: \(0.532502\)
Root analytic conductor: \(0.729727\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1067} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1067,\ (\ :0),\ -0.523 - 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3983532344\)
\(L(\frac12)\) \(\approx\) \(0.3983532344\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.608 - 0.793i)T \)
97 \( 1 + (-0.793 - 0.608i)T \)
good2 \( 1 + (-0.258 - 0.965i)T^{2} \)
3 \( 1 + (1.91 - 0.252i)T + (0.965 - 0.258i)T^{2} \)
5 \( 1 + (0.583 + 0.665i)T + (-0.130 + 0.991i)T^{2} \)
7 \( 1 + (0.793 - 0.608i)T^{2} \)
13 \( 1 + (-0.130 + 0.991i)T^{2} \)
17 \( 1 + (-0.793 - 0.608i)T^{2} \)
19 \( 1 + (-0.923 + 0.382i)T^{2} \)
23 \( 1 + (1.34 - 0.665i)T + (0.608 - 0.793i)T^{2} \)
29 \( 1 + (0.991 + 0.130i)T^{2} \)
31 \( 1 + (-0.241 - 1.83i)T + (-0.965 + 0.258i)T^{2} \)
37 \( 1 + (0.837 - 1.69i)T + (-0.608 - 0.793i)T^{2} \)
41 \( 1 + (-0.991 - 0.130i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
53 \( 1 + (0.158 - 0.207i)T + (-0.258 - 0.965i)T^{2} \)
59 \( 1 + (1.60 + 0.793i)T + (0.608 + 0.793i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.389 - 1.95i)T + (-0.923 + 0.382i)T^{2} \)
71 \( 1 + (-0.641 + 0.0420i)T + (0.991 - 0.130i)T^{2} \)
73 \( 1 + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (0.793 + 0.608i)T^{2} \)
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

−10.48970832356248602723827029798, −9.779598415177131522765325394831, −8.683028722100993202608863378925, −7.69767622272313164233063121785, −6.87931395923671393649565159795, −6.25363408755655974220062030252, −5.02989682934968707059142222053, −4.45064811805642389879154917211, −3.62335984153133250977262623178, −1.52898247956457764245526933805, 0.48722567172832297592643217364, 1.93418041285458702402076523709, 3.85927334219657456564248780013, 4.80130603105633559417745539836, 5.97370126188045452606222540556, 6.08224663551071017512432328653, 7.02229543423569831500577936472, 7.77269269607911768122853085310, 9.323421236769868842396838820479, 10.18850922347195276833761163095

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