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Properties

Label 2-1067-1067.461-c0-0-0
Degree $2$
Conductor $1067$
Sign $-0.833 + 0.552i$
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  + (−0.448 − 1.67i)3-s + (0.866 − 0.5i)4-s + (−0.158 − 0.207i)5-s + (−1.73 + 1.00i)9-s + (−0.965 − 0.258i)11-s + (−1.22 − 1.22i)12-s + (−0.275 + 0.358i)15-s + (0.499 − 0.866i)16-s + (−0.241 − 0.0999i)20-s + (−0.158 − 1.20i)23-s + (0.241 − 0.900i)25-s + (1.22 + 1.22i)27-s + (0.366 + 1.36i)31-s + 1.73i·33-s + (−0.999 + 1.73i)36-s + (−0.258 + 1.96i)37-s + ⋯
L(s)  = 1  + (−0.448 − 1.67i)3-s + (0.866 − 0.5i)4-s + (−0.158 − 0.207i)5-s + (−1.73 + 1.00i)9-s + (−0.965 − 0.258i)11-s + (−1.22 − 1.22i)12-s + (−0.275 + 0.358i)15-s + (0.499 − 0.866i)16-s + (−0.241 − 0.0999i)20-s + (−0.158 − 1.20i)23-s + (0.241 − 0.900i)25-s + (1.22 + 1.22i)27-s + (0.366 + 1.36i)31-s + 1.73i·33-s + (−0.999 + 1.73i)36-s + (−0.258 + 1.96i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1067\)    =    \(11 \cdot 97\)
Sign: $-0.833 + 0.552i$
Analytic conductor: \(0.532502\)
Root analytic conductor: \(0.729727\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1067} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1067,\ (\ :0),\ -0.833 + 0.552i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9256779582\)
\(L(\frac12)\) \(\approx\) \(0.9256779582\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.965 + 0.258i)T \)
97 \( 1 + (-0.965 + 0.258i)T \)
good2 \( 1 + (-0.866 + 0.5i)T^{2} \)
3 \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \)
5 \( 1 + (0.158 + 0.207i)T + (-0.258 + 0.965i)T^{2} \)
7 \( 1 + (-0.965 - 0.258i)T^{2} \)
13 \( 1 + (0.258 - 0.965i)T^{2} \)
17 \( 1 + (-0.965 + 0.258i)T^{2} \)
19 \( 1 + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (0.158 + 1.20i)T + (-0.965 + 0.258i)T^{2} \)
29 \( 1 + (-0.258 + 0.965i)T^{2} \)
31 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
37 \( 1 + (0.258 - 1.96i)T + (-0.965 - 0.258i)T^{2} \)
41 \( 1 + (-0.258 + 0.965i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (-0.0340 + 0.258i)T + (-0.965 - 0.258i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.607 + 1.46i)T + (-0.707 - 0.707i)T^{2} \)
71 \( 1 + (-1.57 - 1.20i)T + (0.258 + 0.965i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 + (-0.965 + 0.258i)T^{2} \)
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

−10.06318446751653812717003050282, −8.426788864724373337366035310115, −8.087867358095247710864915720758, −6.95024171609762745689161577960, −6.63543773880348345387636942447, −5.71523640698222251615386802683, −4.91262531783963050166196808307, −2.92341561509029908897837127924, −2.10996629743935343801565862601, −0.882470200517653831019718899398, 2.41603411751349502000563616904, 3.46073125074010920755661168126, 4.16111182286584662159713087154, 5.34277478702966808628893916276, 5.90030809738707152964591942974, 7.19567115647426977413373130373, 7.88259262688971630889853166603, 9.005945604339506857285302433060, 9.768934500162549732982580907156, 10.53954805645719581244586473871

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