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Properties

Label 1-1860-1860.1739-r1-0-0
Degree $1$
Conductor $1860$
Sign $-0.978 - 0.205i$
Analytic cond. $199.884$
Root an. cond. $199.884$
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  + (0.913 − 0.406i)7-s + (0.104 − 0.994i)11-s + (0.669 − 0.743i)13-s + (0.104 + 0.994i)17-s + (−0.669 − 0.743i)19-s + (−0.809 + 0.587i)23-s + (0.309 − 0.951i)29-s + (−0.5 + 0.866i)37-s + (0.978 − 0.207i)41-s + (−0.669 − 0.743i)43-s + (−0.309 − 0.951i)47-s + (0.669 − 0.743i)49-s + (−0.913 − 0.406i)53-s + (−0.978 − 0.207i)59-s − 61-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)7-s + (0.104 − 0.994i)11-s + (0.669 − 0.743i)13-s + (0.104 + 0.994i)17-s + (−0.669 − 0.743i)19-s + (−0.809 + 0.587i)23-s + (0.309 − 0.951i)29-s + (−0.5 + 0.866i)37-s + (0.978 − 0.207i)41-s + (−0.669 − 0.743i)43-s + (−0.309 − 0.951i)47-s + (0.669 − 0.743i)49-s + (−0.913 − 0.406i)53-s + (−0.978 − 0.207i)59-s − 61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.978 - 0.205i$
Analytic conductor: \(199.884\)
Root analytic conductor: \(199.884\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (1739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1860,\ (1:\ ),\ -0.978 - 0.205i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1125302894 - 1.085983901i\)
\(L(\frac12)\) \(\approx\) \(0.1125302894 - 1.085983901i\)
\(L(1)\) \(\approx\) \(1.033305814 - 0.2477745979i\)
\(L(1)\) \(\approx\) \(1.033305814 - 0.2477745979i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
31 \( 1 \)
good7 \( 1 + (0.913 - 0.406i)T \)
11 \( 1 + (0.104 - 0.994i)T \)
13 \( 1 + (0.669 - 0.743i)T \)
17 \( 1 + (0.104 + 0.994i)T \)
19 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.978 - 0.207i)T \)
43 \( 1 + (-0.669 - 0.743i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.913 - 0.406i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.913 + 0.406i)T \)
73 \( 1 + (-0.104 + 0.994i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.39520783647653108828682699421, −19.60432429151368019230990676708, −18.57428449753458460208338883039, −18.16861306524613929507624113441, −17.50051288754451661379334752177, −16.52354420404379050435369408245, −15.941205292746376482359165751867, −15.02929133059508518623788583979, −14.32318132188729895082839639733, −13.90235425484554065196201972269, −12.59787203057265409458713200586, −12.21510811445838786002968643522, −11.30175841820556792022250300007, −10.68301984380269682171505771651, −9.67158542880799027687954685845, −8.98541492021025278596980525551, −8.167266040248674593048647663166, −7.42937921495492364599903991812, −6.5240392401084705975683254992, −5.71335572842369059437888717419, −4.65198973673513784028199569797, −4.25976212360176744224204784793, −2.97459360527401035889560417053, −1.953660903412082405900355633713, −1.361948166253357187422972592549, 0.187846266280859662950943239985, 1.17426332147773927923145072396, 2.05527159505901159476626317895, 3.27111268755394079165863746550, 3.97662482182835747174431073336, 4.89411757987836954891178312809, 5.83462370045812804934320200589, 6.43203673296235412896764483003, 7.62091102136946928789133774225, 8.27137874375615061821693550791, 8.75260031762362520147949766659, 9.99289353173002285637439014531, 10.71760985018925337634834156547, 11.25356217095447120761286687657, 12.04842961820079300933832276562, 13.09700740080195250398575190065, 13.66331419893570372723736217414, 14.34723163714069932763980442702, 15.29228624326386314838441425106, 15.74338547622928908344854195513, 16.922486119162434829760612871, 17.281821449550132605454780895278, 18.1008473754296942450091710100, 18.84112493427086993546642031257, 19.6809372934510496838859862960

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