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Properties

Label 2-860-860.239-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: $\chi_{860} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 860,\ (\ :0),\ -0.736 + 0.675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9908861184\)
\(L(\frac12)\) \(\approx\) \(0.9908861184\)
\(L(1)\) 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

−10.34237648726650306653265877239, −9.512587736781481374786357253900, −8.725924207561523669059277722352, −7.20517361272961243379755415293, −6.19990701372162772645100156270, −5.57179949901841852260005075329, −4.70572661938393289100181624650, −3.72773915153050512239087083157, −2.43881464933894922051080562381, −0.903918708845000453669848585957, 2.48028327108556910088495356129, 3.30960808146773401832587090870, 4.89787198773474639865667796949, 5.67936647718795897665526866486, 6.25977247255706004592019163472, 6.79236920257128848449958804607, 8.042284551663517110390170644010, 9.043800799530601624304553025303, 9.637284151930886770313264496007, 10.91314123271384493770064951887

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