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Properties

Label 1-40-40.29-r0-0-0
Degree 11
Conductor 4040
Sign 11
Analytic cond. 0.1857590.185759
Root an. cond. 0.1857590.185759
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

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

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

Invariants

Degree: 11
Conductor: 4040    =    23⋅52^{3} \cdot 5
Sign: 11
Analytic conductor: 0.1857590.185759
Root analytic conductor: 0.1857590.185759
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ40(29,⋅)\chi_{40} (29, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (1, 40, (0: ), 1)(1,\ 40,\ (0:\ ),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.97248885050.9724888505
L(12)L(\frac12) ≈\approx 0.97248885050.9724888505
L(1)L(1) ≈\approx 1.1500865221.150086522
L(1)L(1) ≈\approx 1.1500865221.150086522

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
good3 1+T 1 + T
7 1−T 1 - T
11 1−T 1 - T
13 1+T 1 + T
17 1−T 1 - T
19 1−T 1 - T
23 1−T 1 - T
29 1−T 1 - T
31 1+T 1 + T
37 1+T 1 + T
41 1+T 1 + T
43 1+T 1 + T
47 1−T 1 - T
53 1+T 1 + T
59 1−T 1 - T
61 1−T 1 - T
67 1+T 1 + T
71 1+T 1 + T
73 1−T 1 - T
79 1+T 1 + T
83 1+T 1 + T
89 1+T 1 + T
97 1−T 1 - T
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   L(s)=∏p (1−αp p−s)−1L(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 ZZ-function along the critical line