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Properties

Label 1-40-40.29-r0-0-0
Degree $1$
Conductor $40$
Sign $1$
Analytic cond. $0.185759$
Root an. cond. $0.185759$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s − 11-s + 13-s − 17-s − 19-s − 21-s − 23-s + 27-s − 29-s + 31-s − 33-s + 37-s + 39-s + 41-s + 43-s − 47-s + 49-s − 51-s + 53-s − 57-s − 59-s − 61-s − 63-s + 67-s − 69-s + ⋯
L(s)  = 1  + 3-s − 7-s + 9-s − 11-s + 13-s − 17-s − 19-s − 21-s − 23-s + 27-s − 29-s + 31-s − 33-s + 37-s + 39-s + 41-s + 43-s − 47-s + 49-s − 51-s + 53-s − 57-s − 59-s − 61-s − 63-s + 67-s − 69-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(0.185759\)
Root analytic conductor: \(0.185759\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{40} (29, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 40,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9724888505\)
\(L(\frac12)\) \(\approx\) \(0.9724888505\)
\(L(1)\) \(\approx\) \(1.150086522\)
\(L(1)\) \(\approx\) \(1.150086522\)

Euler product

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

−35.46871815933931066557067894264, −33.77397195045299345934334946200, −32.48992182986535727451601240753, −31.65702877158575683499667689923, −30.53574562468521890003722781377, −29.2753109321509594977720938410, −27.96504990212936504067312446763, −26.2614027105047778869654803926, −25.87097298765072715474022116727, −24.43655444208668619202989797271, −23.088687094180323858461465199341, −21.56472815482081368499382900513, −20.40160148601184344649904072245, −19.28135808746821972149080789134, −18.18218004680212107005435911139, −16.16663433364409385216294347220, −15.2714728878515316883170383332, −13.60430796560710377807192393179, −12.81517863613023536936946170700, −10.63827697953891807504773945162, −9.28044435522920809581999772668, −7.98779634495761720535224006097, −6.33545319091356197519845871198, −4.03159701418841483370658989241, −2.48821020930540853564011047706, 2.48821020930540853564011047706, 4.03159701418841483370658989241, 6.33545319091356197519845871198, 7.98779634495761720535224006097, 9.28044435522920809581999772668, 10.63827697953891807504773945162, 12.81517863613023536936946170700, 13.60430796560710377807192393179, 15.2714728878515316883170383332, 16.16663433364409385216294347220, 18.18218004680212107005435911139, 19.28135808746821972149080789134, 20.40160148601184344649904072245, 21.56472815482081368499382900513, 23.088687094180323858461465199341, 24.43655444208668619202989797271, 25.87097298765072715474022116727, 26.2614027105047778869654803926, 27.96504990212936504067312446763, 29.2753109321509594977720938410, 30.53574562468521890003722781377, 31.65702877158575683499667689923, 32.48992182986535727451601240753, 33.77397195045299345934334946200, 35.46871815933931066557067894264

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