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Properties

Label 2-1067-1067.681-c0-0-0
Degree $2$
Conductor $1067$
Sign $0.308 - 0.951i$
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 + (0.665 + 1.34i)5-s + (−0.189 + 0.707i)9-s + (−0.130 − 0.991i)11-s + (−0.478 + 0.198i)12-s + (−0.698 − 0.344i)15-s + (0.866 + 0.499i)16-s + (0.293 + 1.47i)20-s + (−1.18 − 1.34i)23-s + (−0.767 + 1.00i)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 + (0.0983 + 0.0862i)37-s + ⋯
L(s)  = 1  + (−0.410 + 0.315i)3-s + (0.965 + 0.258i)4-s + (0.665 + 1.34i)5-s + (−0.189 + 0.707i)9-s + (−0.130 − 0.991i)11-s + (−0.478 + 0.198i)12-s + (−0.698 − 0.344i)15-s + (0.866 + 0.499i)16-s + (0.293 + 1.47i)20-s + (−1.18 − 1.34i)23-s + (−0.767 + 1.00i)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 + (0.0983 + 0.0862i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1067\)    =    \(11 \cdot 97\)
Sign: $0.308 - 0.951i$
Analytic conductor: \(0.532502\)
Root analytic conductor: \(0.729727\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1067} (681, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1067,\ (\ :0),\ 0.308 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.204917232\)
\(L(\frac12)\) \(\approx\) \(1.204917232\)
\(L(1)\) 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 \( 1 + (-0.665 - 1.34i)T + (-0.608 + 0.793i)T^{2} \)
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 \( 1 + (1.18 + 1.34i)T + (-0.130 + 0.991i)T^{2} \)
29 \( 1 + (0.793 + 0.608i)T^{2} \)
31 \( 1 + (0.465 + 0.607i)T + (-0.258 + 0.965i)T^{2} \)
37 \( 1 + (-0.0983 - 0.0862i)T + (0.130 + 0.991i)T^{2} \)
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 \( 1 + (0.869 - 0.991i)T + (-0.130 - 0.991i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.57 - 1.05i)T + (0.382 + 0.923i)T^{2} \)
71 \( 1 + (-1.88 + 0.641i)T + (0.793 - 0.608i)T^{2} \)
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

−10.45380001371770723612163586070, −9.834666142574791018442992719543, −8.410437949910000476942818472890, −7.71354387435317689300307617706, −6.68612427429077593130445086102, −6.14703699773596388431557928923, −5.42698506389916229780727067058, −3.90997870486005752011527872466, −2.79402747935443441209340765618, −2.15701465607510133754016963113, 1.28629528881719978592292595901, 2.12547522138973401958783025083, 3.70758133417618617821232739416, 5.05842419301597867417898419688, 5.65979440009865465628602533797, 6.45951665967265853564704726601, 7.31712962202594049896171584747, 8.222021181446643724287478507300, 9.393637586347149914189366873502, 9.715888251913268029963682549944

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