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Properties

Label 2-10944-1.1-c1-0-58
Degree 22
Conductor 1094410944
Sign −1-1
Analytic cond. 87.388287.3882
Root an. cond. 9.348169.34816
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 3·7-s + 3·11-s + 6·13-s − 3·17-s − 19-s + 4·23-s − 4·25-s − 10·29-s − 2·31-s − 3·35-s − 8·37-s + 8·41-s − 43-s + 3·47-s + 2·49-s − 6·53-s + 3·55-s − 7·61-s + 6·65-s + 8·67-s + 12·71-s − 11·73-s − 9·77-s − 4·83-s − 3·85-s − 10·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.13·7-s + 0.904·11-s + 1.66·13-s − 0.727·17-s − 0.229·19-s + 0.834·23-s − 4/5·25-s − 1.85·29-s − 0.359·31-s − 0.507·35-s − 1.31·37-s + 1.24·41-s − 0.152·43-s + 0.437·47-s + 2/7·49-s − 0.824·53-s + 0.404·55-s − 0.896·61-s + 0.744·65-s + 0.977·67-s + 1.42·71-s − 1.28·73-s − 1.02·77-s − 0.439·83-s − 0.325·85-s − 1.05·89-s + ⋯

Functional equation

Λ(s)=(10944s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 10944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(10944s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1094410944    =    26⋅32⋅192^{6} \cdot 3^{2} \cdot 19
Sign: −1-1
Analytic conductor: 87.388287.3882
Root analytic conductor: 9.348169.34816
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 10944, ( :1/2), −1)(2,\ 10944,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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
3 1 1
19 1+T 1 + T
good5 1−T+pT2 1 - T + p T^{2}
7 1+3T+pT2 1 + 3 T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+10T+pT2 1 + 10 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1−8T+pT2 1 - 8 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1−3T+pT2 1 - 3 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+7T+pT2 1 + 7 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+11T+pT2 1 + 11 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1+2T+pT2 1 + 2 T + p T^{2}
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   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

−16.81887350203953, −16.18180407078120, −15.69769146875137, −15.22877781182094, −14.44533350536324, −13.81724510311056, −13.33926301942935, −12.89769677467479, −12.36199706404542, −11.40510641695561, −11.05769589983429, −10.47064103034283, −9.491841424759626, −9.297791790124632, −8.748240991447002, −7.937722605290241, −6.964972819076286, −6.611738286011658, −5.914911053835027, −5.504292221833288, −4.257775732469895, −3.738231292327022, −3.139452097998629, −2.035993201395619, −1.279511364527950, 0, 1.279511364527950, 2.035993201395619, 3.139452097998629, 3.738231292327022, 4.257775732469895, 5.504292221833288, 5.914911053835027, 6.611738286011658, 6.964972819076286, 7.937722605290241, 8.748240991447002, 9.297791790124632, 9.491841424759626, 10.47064103034283, 11.05769589983429, 11.40510641695561, 12.36199706404542, 12.89769677467479, 13.33926301942935, 13.81724510311056, 14.44533350536324, 15.22877781182094, 15.69769146875137, 16.18180407078120, 16.81887350203953

Graph of the ZZ-function along the critical line