↑

Properties

Label 2-1160-1.1-c3-0-53
Degree 22
Conductor 11601160
Sign −1-1
Analytic cond. 68.442268.4422
Root an. cond. 8.272988.27298
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8.44·3-s + 5·5-s + 16.2·7-s + 44.3·9-s − 0.0653·11-s − 34.3·13-s − 42.2·15-s − 71.6·17-s + 34.2·19-s − 136.·21-s + 68.5·23-s + 25·25-s − 146.·27-s − 29·29-s − 280.·31-s + 0.552·33-s + 81.0·35-s + 403.·37-s + 289.·39-s − 318.·41-s − 273.·43-s + 221.·45-s + 518.·47-s − 80.1·49-s + 605.·51-s − 492.·53-s − 0.326·55-s + ⋯
L(s)  = 1  − 1.62·3-s + 0.447·5-s + 0.875·7-s + 1.64·9-s − 0.00179·11-s − 0.732·13-s − 0.726·15-s − 1.02·17-s + 0.413·19-s − 1.42·21-s + 0.621·23-s + 0.200·25-s − 1.04·27-s − 0.185·29-s − 1.62·31-s + 0.00291·33-s + 0.391·35-s + 1.79·37-s + 1.19·39-s − 1.21·41-s − 0.969·43-s + 0.734·45-s + 1.60·47-s − 0.233·49-s + 1.66·51-s − 1.27·53-s − 0.000801·55-s + ⋯

Functional equation

Λ(s)=(1160s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1160s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11601160    =    23⋅5⋅292^{3} \cdot 5 \cdot 29
Sign: −1-1
Analytic conductor: 68.442268.4422
Root analytic conductor: 8.272988.27298
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1160, ( :3/2), −1)(2,\ 1160,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

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 1−5T 1 - 5T
29 1+29T 1 + 29T
good3 1+8.44T+27T2 1 + 8.44T + 27T^{2}
7 1−16.2T+343T2 1 - 16.2T + 343T^{2}
11 1+0.0653T+1.33e3T2 1 + 0.0653T + 1.33e3T^{2}
13 1+34.3T+2.19e3T2 1 + 34.3T + 2.19e3T^{2}
17 1+71.6T+4.91e3T2 1 + 71.6T + 4.91e3T^{2}
19 1−34.2T+6.85e3T2 1 - 34.2T + 6.85e3T^{2}
23 1−68.5T+1.21e4T2 1 - 68.5T + 1.21e4T^{2}
31 1+280.T+2.97e4T2 1 + 280.T + 2.97e4T^{2}
37 1−403.T+5.06e4T2 1 - 403.T + 5.06e4T^{2}
41 1+318.T+6.89e4T2 1 + 318.T + 6.89e4T^{2}
43 1+273.T+7.95e4T2 1 + 273.T + 7.95e4T^{2}
47 1−518.T+1.03e5T2 1 - 518.T + 1.03e5T^{2}
53 1+492.T+1.48e5T2 1 + 492.T + 1.48e5T^{2}
59 1−861.T+2.05e5T2 1 - 861.T + 2.05e5T^{2}
61 1−125.T+2.26e5T2 1 - 125.T + 2.26e5T^{2}
67 1−1.03e3T+3.00e5T2 1 - 1.03e3T + 3.00e5T^{2}
71 1+91.9T+3.57e5T2 1 + 91.9T + 3.57e5T^{2}
73 1−936.T+3.89e5T2 1 - 936.T + 3.89e5T^{2}
79 1+768.T+4.93e5T2 1 + 768.T + 4.93e5T^{2}
83 1−666.T+5.71e5T2 1 - 666.T + 5.71e5T^{2}
89 1−736.T+7.04e5T2 1 - 736.T + 7.04e5T^{2}
97 1+1.34e3T+9.12e5T2 1 + 1.34e3T + 9.12e5T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.213782719806049952217996588694, −8.057866623359042349468440214382, −7.06327354768894779708528907706, −6.47323258162652762101611186582, −5.30884823049643253771071556349, −5.11988914680641522474192705937, −4.04856419690327727213677573137, −2.31540376413599944374847371773, −1.20320400037759451762651098439, 0, 1.20320400037759451762651098439, 2.31540376413599944374847371773, 4.04856419690327727213677573137, 5.11988914680641522474192705937, 5.30884823049643253771071556349, 6.47323258162652762101611186582, 7.06327354768894779708528907706, 8.057866623359042349468440214382, 9.213782719806049952217996588694

Graph of the ZZ-function along the critical line