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Properties

Label 1-40-40.13-r1-0-0
Degree $1$
Conductor $40$
Sign $-0.525 - 0.850i$
Analytic cond. $4.29859$
Root an. cond. $4.29859$
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  − i·3-s − i·7-s − 9-s − 11-s − i·13-s − i·17-s + 19-s − 21-s + i·23-s + i·27-s + 29-s + 31-s + i·33-s + i·37-s − 39-s + ⋯
L(s)  = 1  − i·3-s − i·7-s − 9-s − 11-s − i·13-s − i·17-s + 19-s − 21-s + i·23-s + i·27-s + 29-s + 31-s + i·33-s + i·37-s − 39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.525 - 0.850i$
Analytic conductor: \(4.29859\)
Root analytic conductor: \(4.29859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 40,\ (1:\ ),\ -0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5840185059 - 1.047498216i\)
\(L(\frac12)\) \(\approx\) \(0.5840185059 - 1.047498216i\)
\(L(1)\) \(\approx\) \(0.8450865538 - 0.5222922137i\)
\(L(1)\) \(\approx\) \(0.8450865538 - 0.5222922137i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + T \)
7 \( 1 - iT \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 - iT \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 - T \)
41 \( 1 \)
43 \( 1 - iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 - iT \)
67 \( 1 \)
71 \( 1 + T \)
73 \( 1 \)
79 \( 1 - T \)
83 \( 1 \)
89 \( 1 + iT \)
97 \( 1 \)
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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

−34.889410722913579019042779215255, −34.00593978863417924291162844420, −32.86231382272903645228213911845, −31.67062597000032905452552710638, −30.87016611756245345183516519694, −28.75483491129007218737184032280, −28.294360278563347705201931249151, −26.74136643289033570614674792378, −25.92992580013884052785017865113, −24.44019338983484927528478730337, −22.93786179582473127180677905639, −21.66654677955369228655182343419, −20.95428797352835588493580628252, −19.35445000819576732156807553178, −17.95182105004652510585130909180, −16.35256477930716606495463617653, −15.44477138266542007247681781931, −14.179957072571212635060024343162, −12.314118211724954003592304568031, −10.909635654089499426674194736919, −9.5392668092564573127924385132, −8.29247179462172609731476339067, −5.98776315237030887203366066327, −4.5862074698134147263949347033, −2.712571261673437121457390540174, 0.78175758422594404739951629151, 2.96145731604313918366439218618, 5.31837883635310062185777995034, 7.097471960447107766621899830927, 8.06518186518042676061520118383, 10.13039025039574289642412239789, 11.633914964105169352606859301659, 13.141235995668734664727808836599, 13.94588662492370748761203109124, 15.79427994969970989519583146281, 17.38576376059781498840809763208, 18.29799312238130571037906726382, 19.74478701066218196150919767223, 20.695607168607225609497625932771, 22.69999016720834375730312406956, 23.53597150240823310138026442521, 24.722920665161044269632960455613, 25.90667840123069701772422442237, 27.174906699397835017340244911219, 28.83127379121171043186545706856, 29.66415484553858424483262837154, 30.720036159678105293485643075963, 31.88487219300669509430517189192, 33.368390162792623722285111405705, 34.512665885422244155468985822560

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