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Properties

Label 2-860-860.99-c0-0-0
Degree $2$
Conductor $860$
Sign $0.941 - 0.336i$
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 + 1.29i)3-s + (−0.900 − 0.433i)4-s + (−0.988 − 0.149i)5-s + (−0.955 − 1.65i)6-s + (0.222 − 0.385i)7-s + (0.623 − 0.781i)8-s + (0.198 − 2.64i)9-s + (0.365 − 0.930i)10-s + (1.82 − 0.563i)12-s + (0.326 + 0.302i)14-s + (1.57 − 1.07i)15-s + (0.623 + 0.781i)16-s + (2.53 + 0.781i)18-s + (0.826 + 0.563i)20-s + (0.189 + 0.829i)21-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−1.40 + 1.29i)3-s + (−0.900 − 0.433i)4-s + (−0.988 − 0.149i)5-s + (−0.955 − 1.65i)6-s + (0.222 − 0.385i)7-s + (0.623 − 0.781i)8-s + (0.198 − 2.64i)9-s + (0.365 − 0.930i)10-s + (1.82 − 0.563i)12-s + (0.326 + 0.302i)14-s + (1.57 − 1.07i)15-s + (0.623 + 0.781i)16-s + (2.53 + 0.781i)18-s + (0.826 + 0.563i)20-s + (0.189 + 0.829i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(860\)    =    \(2^{2} \cdot 5 \cdot 43\)
Sign: $0.941 - 0.336i$
Analytic conductor: \(0.429195\)
Root analytic conductor: \(0.655130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{860} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 860,\ (\ :0),\ 0.941 - 0.336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2937453672\)
\(L(\frac12)\) \(\approx\) \(0.2937453672\)
\(L(1)\) 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 \( 1 + (0.988 + 0.149i)T \)
43 \( 1 + (-0.365 + 0.930i)T \)
good3 \( 1 + (1.40 - 1.29i)T + (0.0747 - 0.997i)T^{2} \)
7 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.623 + 0.781i)T^{2} \)
13 \( 1 + (0.733 - 0.680i)T^{2} \)
17 \( 1 + (-0.955 + 0.294i)T^{2} \)
19 \( 1 + (0.988 - 0.149i)T^{2} \)
23 \( 1 + (1.48 + 1.01i)T + (0.365 + 0.930i)T^{2} \)
29 \( 1 + (0.535 + 0.496i)T + (0.0747 + 0.997i)T^{2} \)
31 \( 1 + (-0.826 + 0.563i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.440 + 1.92i)T + (-0.900 - 0.433i)T^{2} \)
47 \( 1 + (-0.900 - 0.433i)T + (0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.733 + 0.680i)T^{2} \)
59 \( 1 + (0.222 - 0.974i)T^{2} \)
61 \( 1 + (-1.19 - 0.367i)T + (0.826 + 0.563i)T^{2} \)
67 \( 1 + (0.147 + 1.97i)T + (-0.988 + 0.149i)T^{2} \)
71 \( 1 + (-0.365 + 0.930i)T^{2} \)
73 \( 1 + (0.733 - 0.680i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1.07 + 0.997i)T + (0.0747 - 0.997i)T^{2} \)
89 \( 1 + (0.109 - 0.101i)T + (0.0747 - 0.997i)T^{2} \)
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

−10.59167059386779053635770620065, −9.581114825363924740117622892880, −8.810500402241806678765324399510, −7.79110114285904059138603513028, −6.86674877493556942840839610753, −5.94939446872541903418280342680, −5.19471995970396955898173621197, −4.18013596623172348904313992493, −3.94362483991944405363005402999, −0.44581868778082234416969175701, 1.22924584790020120435310953857, 2.44670332525210685146141515288, 3.97281923024145752201115601920, 5.05087832415910682682087769308, 5.89734722965671081860039994988, 7.05969068888756733829820865567, 7.81924760531896656091344086114, 8.385380816564417092329456262365, 9.773669517160865497085848360687, 10.75243845864278604858339940715

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