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Properties

Label 2-1067-1067.967-c0-0-0
Degree $2$
Conductor $1067$
Sign $0.645 + 0.763i$
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 + (−1.34 − 1.18i)5-s + (2.63 + 0.707i)9-s + (−0.608 + 0.793i)11-s + (0.739 − 1.78i)12-s + (−2.28 − 2.60i)15-s + (−0.866 − 0.499i)16-s + (−1.49 + 0.996i)20-s + (−0.583 + 1.18i)23-s + (0.289 + 2.19i)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 + (−0.576 + 0.284i)37-s + ⋯
L(s)  = 1  + (1.91 + 0.252i)3-s + (0.258 − 0.965i)4-s + (−1.34 − 1.18i)5-s + (2.63 + 0.707i)9-s + (−0.608 + 0.793i)11-s + (0.739 − 1.78i)12-s + (−2.28 − 2.60i)15-s + (−0.866 − 0.499i)16-s + (−1.49 + 0.996i)20-s + (−0.583 + 1.18i)23-s + (0.289 + 2.19i)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 + (−0.576 + 0.284i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1067\)    =    \(11 \cdot 97\)
Sign: $0.645 + 0.763i$
Analytic conductor: \(0.532502\)
Root analytic conductor: \(0.729727\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1067} (967, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1067,\ (\ :0),\ 0.645 + 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.609286885\)
\(L(\frac12)\) \(\approx\) \(1.609286885\)
\(L(1)\) 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 \( 1 + (1.34 + 1.18i)T + (0.130 + 0.991i)T^{2} \)
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 \( 1 + (0.583 - 1.18i)T + (-0.608 - 0.793i)T^{2} \)
29 \( 1 + (-0.991 + 0.130i)T^{2} \)
31 \( 1 + (-0.241 + 1.83i)T + (-0.965 - 0.258i)T^{2} \)
37 \( 1 + (0.576 - 0.284i)T + (0.608 - 0.793i)T^{2} \)
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 \( 1 + (0.391 + 0.793i)T + (-0.608 + 0.793i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.128 - 0.0255i)T + (0.923 + 0.382i)T^{2} \)
71 \( 1 + (0.123 - 1.88i)T + (-0.991 - 0.130i)T^{2} \)
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

−9.577805462316718829561278316368, −9.293234309076091395471768371819, −8.239577605720179428903427063993, −7.77670527569696317177247623716, −7.09096523130195066540100530337, −5.39471392453226237292354786487, −4.44332450113905135656572718697, −3.90099951547377587585196898119, −2.58273208519936604419061499253, −1.46953938802172559578475646391, 2.35386346360254385633552007746, 3.07001348974088496020567114450, 3.57444330808857479783646092517, 4.38227843387374240527153205789, 6.61619429050240241163351109474, 7.18523723290685937282796264974, 7.84051172281633126981714843513, 8.398999837296005882053138974561, 8.877737357325723079353455358278, 10.34084826290535775532700712971

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