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Properties

Label 2-2016-1.1-c1-0-22
Degree 22
Conductor 20162016
Sign −1-1
Analytic cond. 16.097816.0978
Root an. cond. 4.012214.01221
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  − 7-s + 2·11-s − 2·13-s − 4·17-s − 4·19-s + 6·23-s − 5·25-s + 2·29-s − 6·37-s − 8·41-s − 8·43-s + 4·47-s + 49-s + 6·53-s − 14·61-s + 4·67-s + 2·71-s − 2·73-s − 2·77-s + 4·79-s − 12·83-s + 2·91-s + 6·97-s − 12·101-s − 8·103-s − 6·107-s − 18·109-s + ⋯
L(s)  = 1  − 0.377·7-s + 0.603·11-s − 0.554·13-s − 0.970·17-s − 0.917·19-s + 1.25·23-s − 25-s + 0.371·29-s − 0.986·37-s − 1.24·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.824·53-s − 1.79·61-s + 0.488·67-s + 0.237·71-s − 0.234·73-s − 0.227·77-s + 0.450·79-s − 1.31·83-s + 0.209·91-s + 0.609·97-s − 1.19·101-s − 0.788·103-s − 0.580·107-s − 1.72·109-s + ⋯

Functional equation

Λ(s)=(2016s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2016s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20162016    =    25⋅32⋅72^{5} \cdot 3^{2} \cdot 7
Sign: −1-1
Analytic conductor: 16.097816.0978
Root analytic conductor: 4.012214.01221
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2016, ( :1/2), −1)(2,\ 2016,\ (\ :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
7 1+T 1 + T
good5 1+pT2 1 + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1+8T+pT2 1 + 8 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+14T+pT2 1 + 14 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−2T+pT2 1 - 2 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1−6T+pT2 1 - 6 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

−8.839220827010344299826518365354, −8.080392159110006302593970355953, −6.90671158809997641288656761731, −6.66151764975017501503964240993, −5.54481409757596501307767728969, −4.64683107259554774725500435991, −3.80855384525275899005212589296, −2.76047872572818436273384792924, −1.67384789151853928239355705819, 0, 1.67384789151853928239355705819, 2.76047872572818436273384792924, 3.80855384525275899005212589296, 4.64683107259554774725500435991, 5.54481409757596501307767728969, 6.66151764975017501503964240993, 6.90671158809997641288656761731, 8.080392159110006302593970355953, 8.839220827010344299826518365354

Graph of the ZZ-function along the critical line