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Properties

Label 2-1067-1067.494-c0-0-0
Degree $2$
Conductor $1067$
Sign $0.781 - 0.624i$
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 + (0.158 + 1.20i)5-s + (1.73 + 1.00i)9-s + (−0.258 − 0.965i)11-s + (−1.22 − 1.22i)12-s + (−0.275 + 2.09i)15-s + (0.499 + 0.866i)16-s + (0.465 − 1.12i)20-s + (0.158 + 0.207i)23-s + (−0.465 + 0.124i)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 + 1.25i)37-s + ⋯
L(s)  = 1  + (1.67 + 0.448i)3-s + (−0.866 − 0.5i)4-s + (0.158 + 1.20i)5-s + (1.73 + 1.00i)9-s + (−0.258 − 0.965i)11-s + (−1.22 − 1.22i)12-s + (−0.275 + 2.09i)15-s + (0.499 + 0.866i)16-s + (0.465 − 1.12i)20-s + (0.158 + 0.207i)23-s + (−0.465 + 0.124i)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 + 1.25i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1067\)    =    \(11 \cdot 97\)
Sign: $0.781 - 0.624i$
Analytic conductor: \(0.532502\)
Root analytic conductor: \(0.729727\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1067} (494, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1067,\ (\ :0),\ 0.781 - 0.624i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.493163468\)
\(L(\frac12)\) \(\approx\) \(1.493163468\)
\(L(1)\) 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 \( 1 + (-0.158 - 1.20i)T + (-0.965 + 0.258i)T^{2} \)
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 \( 1 + (-0.158 - 0.207i)T + (-0.258 + 0.965i)T^{2} \)
29 \( 1 + (-0.965 + 0.258i)T^{2} \)
31 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
37 \( 1 + (0.965 - 1.25i)T + (-0.258 - 0.965i)T^{2} \)
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 \( 1 + (-0.741 + 0.965i)T + (-0.258 - 0.965i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.83 + 0.758i)T + (0.707 + 0.707i)T^{2} \)
71 \( 1 + (1.57 + 0.207i)T + (0.965 + 0.258i)T^{2} \)
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

−10.20496918738099300586501112277, −9.235008935153521727602691925799, −8.703199415557267792420216688571, −7.936546647132653544916833940162, −7.02448850528150653856811884892, −5.90708938746030893536555852801, −4.80760317322212989369867880661, −3.60889156838596525322350862290, −3.18620621252827388463070522900, −1.96051684641420376105962544058, 1.47906113339406307182752258798, 2.65700031249577501744328531457, 3.81574025165678855232698251181, 4.51368540891060007961109237205, 5.48484532722930240875245270134, 7.23592532276538624298986639254, 7.61454291321375109421719389140, 8.677181841926608499522899771142, 8.906248301649001488616172377816, 9.515690335682178046434235402799

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