↑

Properties

Label 2-1860-1860.1139-c0-0-1
Degree $2$
Conductor $1860$
Sign $0.965 - 0.258i$
Analytic cond. $0.928260$
Root an. cond. $0.963462$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s − 5-s + (0.809 − 0.587i)6-s + (−0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.309 − 0.951i)10-s + (0.809 + 0.587i)12-s + (0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)18-s + (1.80 + 0.587i)19-s + (0.809 − 0.587i)20-s + (1.30 − 0.951i)23-s + (−0.309 + 0.951i)24-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s − 5-s + (0.809 − 0.587i)6-s + (−0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.309 − 0.951i)10-s + (0.809 + 0.587i)12-s + (0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)18-s + (1.80 + 0.587i)19-s + (0.809 − 0.587i)20-s + (1.30 − 0.951i)23-s + (−0.309 + 0.951i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.965 - 0.258i$
Analytic conductor: \(0.928260\)
Root analytic conductor: \(0.963462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :0),\ 0.965 - 0.258i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8805846308\)
\(L(\frac12)\) \(\approx\) \(0.8805846308\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + T \)
31 \( 1 + (-0.309 + 0.951i)T \)
good7 \( 1 + (0.309 + 0.951i)T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
29 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (0.809 + 0.587i)T^{2} \)
47 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T^{2} \)
61 \( 1 - 1.90iT - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.062983322650728358893269639191, −8.356641756810136710418914793250, −7.67243500522977328739916440368, −7.16888996607806039546974706033, −6.46186670530347118588049342567, −5.48750284093002915260974169871, −4.83404655929998743129222863774, −3.70491334451218341265604651568, −2.79535548866803986971743982040, −0.852103272735277286414273935796, 1.03902734516361877238058269078, 3.06760707477638920498439212994, 3.30891223862349533646683871086, 4.48428233879645075623191701894, 4.97655005481667650370630086224, 5.81127512649439998327069896776, 7.05233650206445582930015938585, 8.001601156851466331759100389695, 9.097496420126415712034167561372, 9.340113297170179774115268220338

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