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Properties

Label 2-860-860.699-c0-0-0
Degree $2$
Conductor $860$
Sign $-0.734 + 0.678i$
Analytic cond. $0.429195$
Root an. cond. $0.655130$
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  + (−0.623 − 0.781i)2-s + (−0.777 + 0.974i)3-s + (−0.222 + 0.974i)4-s + (−0.900 + 0.433i)5-s + 1.24·6-s − 1.24·7-s + (0.900 − 0.433i)8-s + (−0.123 − 0.541i)9-s + (0.900 + 0.433i)10-s + (−0.777 − 0.974i)12-s + (0.777 + 0.974i)14-s + (0.277 − 1.21i)15-s + (−0.900 − 0.433i)16-s + (−0.346 + 0.433i)18-s + (−0.222 − 0.974i)20-s + (0.969 − 1.21i)21-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)2-s + (−0.777 + 0.974i)3-s + (−0.222 + 0.974i)4-s + (−0.900 + 0.433i)5-s + 1.24·6-s − 1.24·7-s + (0.900 − 0.433i)8-s + (−0.123 − 0.541i)9-s + (0.900 + 0.433i)10-s + (−0.777 − 0.974i)12-s + (0.777 + 0.974i)14-s + (0.277 − 1.21i)15-s + (−0.900 − 0.433i)16-s + (−0.346 + 0.433i)18-s + (−0.222 − 0.974i)20-s + (0.969 − 1.21i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(860\)    =    \(2^{2} \cdot 5 \cdot 43\)
Sign: $-0.734 + 0.678i$
Analytic conductor: \(0.429195\)
Root analytic conductor: \(0.655130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{860} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 860,\ (\ :0),\ -0.734 + 0.678i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06063098714\)
\(L(\frac12)\) \(\approx\) \(0.06063098714\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (0.900 - 0.433i)T \)
43 \( 1 + (-0.900 - 0.433i)T \)
good3 \( 1 + (0.777 - 0.974i)T + (-0.222 - 0.974i)T^{2} \)
7 \( 1 + 1.24T + T^{2} \)
11 \( 1 + (0.900 - 0.433i)T^{2} \)
13 \( 1 + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + (-0.623 - 0.781i)T^{2} \)
19 \( 1 + (0.900 + 0.433i)T^{2} \)
23 \( 1 + (0.0990 + 0.433i)T + (-0.900 + 0.433i)T^{2} \)
29 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.222 - 0.974i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.445 + 1.94i)T + (-0.900 - 0.433i)T^{2} \)
53 \( 1 + (-0.623 - 0.781i)T^{2} \)
59 \( 1 + (-0.623 - 0.781i)T^{2} \)
61 \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \)
67 \( 1 + (0.400 - 1.75i)T + (-0.900 - 0.433i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (-0.623 + 0.781i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.777 - 0.974i)T + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
97 \( 1 + (0.900 - 0.433i)T^{2} \)
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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

−10.23826553259001978039984901232, −9.526422264563766876066925863823, −8.641528200717722368996055428360, −7.56587267778549667696104059153, −6.73882311700845683914540701456, −5.56966608276517438329517870944, −4.19769068686820763331981753675, −3.75265712474946883916971962577, −2.59798006208661820540844699900, −0.086397715983590952712034998250, 1.38276655898392251200632236431, 3.41610799448223418796152422536, 4.80180701012690267919016826474, 5.84439060966475974983781262323, 6.51810071729899591121155159065, 7.28529194610878622090062065678, 7.84941379914786315896969171833, 9.003639997807687207962540267764, 9.570891944245491161852952223427, 10.75982970450184286816392942509

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