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Properties

Label 2-860-860.539-c0-0-1
Degree $2$
Conductor $860$
Sign $-0.431 + 0.901i$
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 + (−0.722 − 1.84i)3-s + (−0.222 + 0.974i)4-s + (0.0747 − 0.997i)5-s + (0.988 − 1.71i)6-s + (−0.623 − 1.07i)7-s + (−0.900 + 0.433i)8-s + (−2.13 + 1.98i)9-s + (0.826 − 0.563i)10-s + (1.95 − 0.294i)12-s + (0.455 − 1.16i)14-s + (−1.88 + 0.582i)15-s + (−0.900 − 0.433i)16-s + (−2.87 − 0.433i)18-s + (0.955 + 0.294i)20-s + (−1.53 + 1.92i)21-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (−0.722 − 1.84i)3-s + (−0.222 + 0.974i)4-s + (0.0747 − 0.997i)5-s + (0.988 − 1.71i)6-s + (−0.623 − 1.07i)7-s + (−0.900 + 0.433i)8-s + (−2.13 + 1.98i)9-s + (0.826 − 0.563i)10-s + (1.95 − 0.294i)12-s + (0.455 − 1.16i)14-s + (−1.88 + 0.582i)15-s + (−0.900 − 0.433i)16-s + (−2.87 − 0.433i)18-s + (0.955 + 0.294i)20-s + (−1.53 + 1.92i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(860\)    =    \(2^{2} \cdot 5 \cdot 43\)
Sign: $-0.431 + 0.901i$
Analytic conductor: \(0.429195\)
Root analytic conductor: \(0.655130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{860} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 860,\ (\ :0),\ -0.431 + 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7781938389\)
\(L(\frac12)\) \(\approx\) \(0.7781938389\)
\(L(1)\) 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 \( 1 + (-0.0747 + 0.997i)T \)
43 \( 1 + (-0.826 + 0.563i)T \)
good3 \( 1 + (0.722 + 1.84i)T + (-0.733 + 0.680i)T^{2} \)
7 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.900 - 0.433i)T^{2} \)
13 \( 1 + (-0.365 - 0.930i)T^{2} \)
17 \( 1 + (0.988 - 0.149i)T^{2} \)
19 \( 1 + (-0.0747 - 0.997i)T^{2} \)
23 \( 1 + (0.425 + 0.131i)T + (0.826 + 0.563i)T^{2} \)
29 \( 1 + (-0.603 + 1.53i)T + (-0.733 - 0.680i)T^{2} \)
31 \( 1 + (-0.955 + 0.294i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.0931 - 0.116i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.222 + 0.974i)T + (-0.900 - 0.433i)T^{2} \)
53 \( 1 + (-0.365 + 0.930i)T^{2} \)
59 \( 1 + (-0.623 - 0.781i)T^{2} \)
61 \( 1 + (-1.78 - 0.268i)T + (0.955 + 0.294i)T^{2} \)
67 \( 1 + (0.109 + 0.101i)T + (0.0747 + 0.997i)T^{2} \)
71 \( 1 + (-0.826 + 0.563i)T^{2} \)
73 \( 1 + (-0.365 - 0.930i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.266 - 0.680i)T + (-0.733 + 0.680i)T^{2} \)
89 \( 1 + (0.535 + 1.36i)T + (-0.733 + 0.680i)T^{2} \)
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

−10.15086123274164423426913015013, −8.790835660441359825016093055501, −7.998084916707620562384719024116, −7.40128793263633256553987240835, −6.59920291766934440014630803367, −5.95694850664718581610134713599, −5.10201453800417519329910769766, −3.97060877611171094697798682877, −2.33637641974289056106423379297, −0.68269130586316330506838704618, 2.68643150815907301793171887506, 3.33500685196048792795422182619, 4.25957195346273776227995669547, 5.31435066366307550744665329752, 5.91074378921279150821253638600, 6.63836657909483982822715872256, 8.690089872473647207221487565128, 9.460393602647371805351263618841, 9.965630855760391023096567410225, 10.73341730545967984281641067527

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