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Properties

Label 2-860-860.619-c0-0-1
Degree $2$
Conductor $860$
Sign $0.821 - 0.570i$
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  + (0.222 + 0.974i)2-s + (1.40 − 0.432i)3-s + (−0.900 + 0.433i)4-s + (0.365 − 0.930i)5-s + (0.733 + 1.26i)6-s + (−0.222 + 0.385i)7-s + (−0.623 − 0.781i)8-s + (0.949 − 0.647i)9-s + (0.988 + 0.149i)10-s + (−1.07 + 0.997i)12-s + (−0.425 − 0.131i)14-s + (0.109 − 1.46i)15-s + (0.623 − 0.781i)16-s + (0.842 + 0.781i)18-s + (0.0747 + 0.997i)20-s + (−0.145 + 0.636i)21-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (1.40 − 0.432i)3-s + (−0.900 + 0.433i)4-s + (0.365 − 0.930i)5-s + (0.733 + 1.26i)6-s + (−0.222 + 0.385i)7-s + (−0.623 − 0.781i)8-s + (0.949 − 0.647i)9-s + (0.988 + 0.149i)10-s + (−1.07 + 0.997i)12-s + (−0.425 − 0.131i)14-s + (0.109 − 1.46i)15-s + (0.623 − 0.781i)16-s + (0.842 + 0.781i)18-s + (0.0747 + 0.997i)20-s + (−0.145 + 0.636i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(860\)    =    \(2^{2} \cdot 5 \cdot 43\)
Sign: $0.821 - 0.570i$
Analytic conductor: \(0.429195\)
Root analytic conductor: \(0.655130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{860} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 860,\ (\ :0),\ 0.821 - 0.570i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.548146290\)
\(L(\frac12)\) \(\approx\) \(1.548146290\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (-0.365 + 0.930i)T \)
43 \( 1 + (-0.988 - 0.149i)T \)
good3 \( 1 + (-1.40 + 0.432i)T + (0.826 - 0.563i)T^{2} \)
7 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.623 - 0.781i)T^{2} \)
13 \( 1 + (-0.955 + 0.294i)T^{2} \)
17 \( 1 + (0.733 - 0.680i)T^{2} \)
19 \( 1 + (-0.365 - 0.930i)T^{2} \)
23 \( 1 + (-0.134 - 1.79i)T + (-0.988 + 0.149i)T^{2} \)
29 \( 1 + (1.88 + 0.582i)T + (0.826 + 0.563i)T^{2} \)
31 \( 1 + (-0.0747 + 0.997i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.162 + 0.712i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.900 - 0.433i)T + (0.623 - 0.781i)T^{2} \)
53 \( 1 + (-0.955 - 0.294i)T^{2} \)
59 \( 1 + (0.222 + 0.974i)T^{2} \)
61 \( 1 + (0.914 + 0.848i)T + (0.0747 + 0.997i)T^{2} \)
67 \( 1 + (0.603 + 0.411i)T + (0.365 + 0.930i)T^{2} \)
71 \( 1 + (0.988 + 0.149i)T^{2} \)
73 \( 1 + (-0.955 + 0.294i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (1.82 - 0.563i)T + (0.826 - 0.563i)T^{2} \)
89 \( 1 + (-1.57 + 0.487i)T + (0.826 - 0.563i)T^{2} \)
97 \( 1 + (-0.623 - 0.781i)T^{2} \)
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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

−9.776644725120148379865169874753, −9.239710445964027664329301665991, −8.786210502714046214732093268524, −7.79270732375357199190510138274, −7.41684596246714805808949712837, −6.06279283780759313333563828161, −5.35188008121230696092055250771, −4.12832402928765549512553979215, −3.18389386223429966303289185339, −1.77756017022405434991753819640, 1.99187429852747176961725041964, 2.86253089994093107075604340291, 3.59380760713489922402954848159, 4.45143515785840793251733359701, 5.80096322696413287362406601840, 6.97838516155398705309656372227, 8.017952399227601729254806455294, 8.936480139128902698883737458837, 9.526556638792035846174247017045, 10.34032371176957760030463435279

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