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Properties

Label 1-1860-1860.1319-r1-0-0
Degree $1$
Conductor $1860$
Sign $0.855 + 0.517i$
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  + (−0.978 − 0.207i)7-s + (−0.669 + 0.743i)11-s + (0.913 + 0.406i)13-s + (−0.669 − 0.743i)17-s + (−0.913 + 0.406i)19-s + (0.309 − 0.951i)23-s + (−0.809 − 0.587i)29-s + (−0.5 + 0.866i)37-s + (0.104 − 0.994i)41-s + (−0.913 + 0.406i)43-s + (0.809 − 0.587i)47-s + (0.913 + 0.406i)49-s + (0.978 − 0.207i)53-s + (−0.104 − 0.994i)59-s − 61-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)7-s + (−0.669 + 0.743i)11-s + (0.913 + 0.406i)13-s + (−0.669 − 0.743i)17-s + (−0.913 + 0.406i)19-s + (0.309 − 0.951i)23-s + (−0.809 − 0.587i)29-s + (−0.5 + 0.866i)37-s + (0.104 − 0.994i)41-s + (−0.913 + 0.406i)43-s + (0.809 − 0.587i)47-s + (0.913 + 0.406i)49-s + (0.978 − 0.207i)53-s + (−0.104 − 0.994i)59-s − 61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.855 + 0.517i$
Analytic conductor: \(199.884\)
Root analytic conductor: \(199.884\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (1319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1860,\ (1:\ ),\ 0.855 + 0.517i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9939472064 + 0.2769993285i\)
\(L(\frac12)\) \(\approx\) \(0.9939472064 + 0.2769993285i\)
\(L(1)\) \(\approx\) \(0.8139644516 + 0.004405048951i\)
\(L(1)\) \(\approx\) \(0.8139644516 + 0.004405048951i\)

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 \( 1 + (-0.978 - 0.207i)T \)
11 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.104 - 0.994i)T \)
43 \( 1 + (-0.913 + 0.406i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.978 - 0.207i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (0.669 - 0.743i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (-0.104 + 0.994i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−19.73843940610825549598877740617, −19.18546177949426437876224724418, −18.47393615045550296876847599507, −17.74938410645919955606235443058, −16.83971056678327519394051358848, −16.16702135011965641983499163408, −15.46695331530222115019156772090, −14.95962820598397693958657042877, −13.66978644225750298764227250731, −13.187380437017891411502046883396, −12.706941014509474908319946834965, −11.59347194035259259305464785982, −10.75741730245841417693273751171, −10.33943800141476005465078437782, −9.07592909168807823011155633940, −8.75440440728428814121414329764, −7.744828274923475081455213441398, −6.8131548554484540898113845801, −5.98857416004639134602425520356, −5.49529754960755537397762668684, −4.208851137248694238758188793, −3.42040600750695736027612850526, −2.69475432623250512694834294478, −1.56035971269194905239556653177, −0.335578080842565972965302291731, 0.49344883245865667996619459082, 1.86911235395598338522660185607, 2.66294710739086658748493679888, 3.6961622649838324872019770029, 4.41992466663454573663454929143, 5.38593343281327450250100190385, 6.44529046676937183059265636105, 6.83970347440012639921966138049, 7.85620910565328669929060140141, 8.7715312337830595706381239725, 9.443180837864948551392353331, 10.338388091807941838701330263996, 10.85101326295345332522920105390, 11.91455369803423350467162015248, 12.655328271749306670341960363631, 13.35120683744071297340171381238, 13.87059792931563963606143019437, 15.10264425570410923556138626581, 15.491139634044654087353728206513, 16.432303680521424832129920247791, 16.87597968431377599626144934142, 17.92372634329130796034357717004, 18.622119646226023879117123171734, 19.11976398585033013595722364824, 20.15309377413786596540870150794

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