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Properties

Label 2-860-860.439-c0-0-1
Degree $2$
Conductor $860$
Sign $0.736 + 0.675i$
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 + 0.108i)3-s + (−0.222 + 0.974i)4-s + (0.826 + 0.563i)5-s + (−0.365 − 0.632i)6-s + (0.623 − 1.07i)7-s + (0.900 − 0.433i)8-s + (−0.445 − 0.137i)9-s + (−0.0747 − 0.997i)10-s + (−0.266 + 0.680i)12-s + (−1.23 + 0.185i)14-s + (0.535 + 0.496i)15-s + (−0.900 − 0.433i)16-s + (0.170 + 0.433i)18-s + (−0.733 + 0.680i)20-s + (0.568 − 0.712i)21-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)2-s + (0.722 + 0.108i)3-s + (−0.222 + 0.974i)4-s + (0.826 + 0.563i)5-s + (−0.365 − 0.632i)6-s + (0.623 − 1.07i)7-s + (0.900 − 0.433i)8-s + (−0.445 − 0.137i)9-s + (−0.0747 − 0.997i)10-s + (−0.266 + 0.680i)12-s + (−1.23 + 0.185i)14-s + (0.535 + 0.496i)15-s + (−0.900 − 0.433i)16-s + (0.170 + 0.433i)18-s + (−0.733 + 0.680i)20-s + (0.568 − 0.712i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(860\)    =    \(2^{2} \cdot 5 \cdot 43\)
Sign: $0.736 + 0.675i$
Analytic conductor: \(0.429195\)
Root analytic conductor: \(0.655130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{860} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 860,\ (\ :0),\ 0.736 + 0.675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.034164705\)
\(L(\frac12)\) \(\approx\) \(1.034164705\)
\(L(1)\) 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 \( 1 + (-0.826 - 0.563i)T \)
43 \( 1 + (0.0747 + 0.997i)T \)
good3 \( 1 + (-0.722 - 0.108i)T + (0.955 + 0.294i)T^{2} \)
7 \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.900 - 0.433i)T^{2} \)
13 \( 1 + (0.988 + 0.149i)T^{2} \)
17 \( 1 + (-0.365 + 0.930i)T^{2} \)
19 \( 1 + (-0.826 + 0.563i)T^{2} \)
23 \( 1 + (0.326 - 0.302i)T + (0.0747 - 0.997i)T^{2} \)
29 \( 1 + (0.147 - 0.0222i)T + (0.955 - 0.294i)T^{2} \)
31 \( 1 + (0.733 + 0.680i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-1.03 - 1.29i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (0.222 - 0.974i)T + (-0.900 - 0.433i)T^{2} \)
53 \( 1 + (0.988 - 0.149i)T^{2} \)
59 \( 1 + (-0.623 - 0.781i)T^{2} \)
61 \( 1 + (0.658 + 1.67i)T + (-0.733 + 0.680i)T^{2} \)
67 \( 1 + (1.57 - 0.487i)T + (0.826 - 0.563i)T^{2} \)
71 \( 1 + (-0.0747 - 0.997i)T^{2} \)
73 \( 1 + (0.988 + 0.149i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (1.95 + 0.294i)T + (0.955 + 0.294i)T^{2} \)
89 \( 1 + (1.88 + 0.284i)T + (0.955 + 0.294i)T^{2} \)
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

−10.18431797261309744759603755292, −9.560477613146797407353964699393, −8.765811676550150396025861019331, −7.86129374575783809137082761870, −7.23424060132947354106288328006, −6.02752332377561792248579004299, −4.58991158661983097574580474857, −3.55255276097083559006401946903, −2.66182304784523456561600305591, −1.53111046392020609127624822629, 1.71376555259076350538037276390, 2.59723643374018159068876439452, 4.51565276461477214473406559731, 5.55867886039334804463664085695, 5.94452535984237991781347537282, 7.26369747976693958434102277663, 8.245067177329755847822924645101, 8.717757237295867974275965040828, 9.254821398331465751684581673056, 10.12374661064584009287070754392

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