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Properties

Label 2-1216-1.1-c1-0-16
Degree 22
Conductor 12161216
Sign −1-1
Analytic cond. 9.709809.70980
Root an. cond. 3.116053.11605
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  − 2·3-s − 3·5-s + 7-s + 9-s + 3·11-s + 4·13-s + 6·15-s − 3·17-s + 19-s − 2·21-s + 4·25-s + 4·27-s − 6·29-s + 4·31-s − 6·33-s − 3·35-s − 2·37-s − 8·39-s − 6·41-s − 43-s − 3·45-s + 3·47-s − 6·49-s + 6·51-s − 12·53-s − 9·55-s − 2·57-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.10·13-s + 1.54·15-s − 0.727·17-s + 0.229·19-s − 0.436·21-s + 4/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s − 1.04·33-s − 0.507·35-s − 0.328·37-s − 1.28·39-s − 0.937·41-s − 0.152·43-s − 0.447·45-s + 0.437·47-s − 6/7·49-s + 0.840·51-s − 1.64·53-s − 1.21·55-s − 0.264·57-s + ⋯

Functional equation

Λ(s)=(1216s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1216s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12161216    =    26⋅192^{6} \cdot 19
Sign: −1-1
Analytic conductor: 9.709809.70980
Root analytic conductor: 3.116053.11605
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1216, ( :1/2), −1)(2,\ 1216,\ (\ :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
19 1−T 1 - T
good3 1+2T+pT2 1 + 2 T + p T^{2}
5 1+3T+pT2 1 + 3 T + p T^{2}
7 1−T+pT2 1 - T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+6T+pT2 1 + 6 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+12T+pT2 1 + 12 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1−T+pT2 1 - T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1+7T+pT2 1 + 7 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1−8T+pT2 1 - 8 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

−9.199158339911279140575946730378, −8.462914393396165009068265832341, −7.65894969899749366452649043261, −6.68601933176938591059893570713, −6.07700564140158959836320975863, −4.96903618429312636799150610379, −4.19306407725294924458030580762, −3.33866387034377157662854676826, −1.39430917521719853589747801849, 0, 1.39430917521719853589747801849, 3.33866387034377157662854676826, 4.19306407725294924458030580762, 4.96903618429312636799150610379, 6.07700564140158959836320975863, 6.68601933176938591059893570713, 7.65894969899749366452649043261, 8.462914393396165009068265832341, 9.199158339911279140575946730378

Graph of the ZZ-function along the critical line