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Properties

Label 1-24-24.11-r0-0-0
Degree 11
Conductor 2424
Sign 11
Analytic cond. 0.1114550.111455
Root an. cond. 0.1114550.111455
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  + 5-s − 7-s − 11-s − 13-s − 17-s + 19-s + 23-s + 25-s + 29-s − 31-s − 35-s − 37-s − 41-s + 43-s + 47-s + 49-s + 53-s − 55-s − 59-s − 61-s − 65-s + 67-s + 71-s + 73-s + 77-s − 79-s − 83-s + ⋯
L(s)  = 1  + 5-s − 7-s − 11-s − 13-s − 17-s + 19-s + 23-s + 25-s + 29-s − 31-s − 35-s − 37-s − 41-s + 43-s + 47-s + 49-s + 53-s − 55-s − 59-s − 61-s − 65-s + 67-s + 71-s + 73-s + 77-s − 79-s − 83-s + ⋯

Functional equation

Λ(s)=(24s/2 ΓR(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(24s/2 ΓR(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 2424    =    23⋅32^{3} \cdot 3
Sign: 11
Analytic conductor: 0.1114550.111455
Root analytic conductor: 0.1114550.111455
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ24(11,⋅)\chi_{24} (11, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (1, 24, (0: ), 1)(1,\ 24,\ (0:\ ),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.70945806140.7094580614
L(12)L(\frac12) ≈\approx 0.70945806140.7094580614
L(1)L(1) ≈\approx 0.93588131010.9358813101
L(1)L(1) ≈\approx 0.93588131010.9358813101

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
3 1 1
good5 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

−38.757045413520358200019297607464, −37.31777594996008237908433584673, −36.33074388995123336856075573360, −34.88719921127767034411870622928, −33.49155789395814591322037091025, −32.42353686755135030793518084391, −31.08725180031803397862031783470, −29.21969083370577778290256668488, −28.86856257427073136901468159166, −26.7561074123782310792383080900, −25.67129733212202220561359782065, −24.40914964223159369359819602153, −22.66785732205079621070293214728, −21.55547824218793740139608072893, −20.034611576750188294726677609, −18.46329885012140044080926945086, −17.09898808691940348689297548911, −15.60365697037025378285194033903, −13.77715270577349586517982430976, −12.631153718166510585493030756588, −10.445721518412924823018776216323, −9.22463743993556115081806332595, −6.97192424379172752833414992011, −5.29243117677719815159648018861, −2.688658132467597586674142549469, 2.688658132467597586674142549469, 5.29243117677719815159648018861, 6.97192424379172752833414992011, 9.22463743993556115081806332595, 10.445721518412924823018776216323, 12.631153718166510585493030756588, 13.77715270577349586517982430976, 15.60365697037025378285194033903, 17.09898808691940348689297548911, 18.46329885012140044080926945086, 20.034611576750188294726677609, 21.55547824218793740139608072893, 22.66785732205079621070293214728, 24.40914964223159369359819602153, 25.67129733212202220561359782065, 26.7561074123782310792383080900, 28.86856257427073136901468159166, 29.21969083370577778290256668488, 31.08725180031803397862031783470, 32.42353686755135030793518084391, 33.49155789395814591322037091025, 34.88719921127767034411870622928, 36.33074388995123336856075573360, 37.31777594996008237908433584673, 38.757045413520358200019297607464

Graph of the ZZ-function along the critical line