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Properties

Label 2-2e7-1.1-c1-0-1
Degree 22
Conductor 128128
Sign 11
Analytic cond. 1.022081.02208
Root an. cond. 1.010981.01098
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·5-s + 4·7-s + 9-s − 2·11-s − 2·13-s − 4·15-s − 2·17-s + 2·19-s + 8·21-s − 4·23-s − 25-s − 4·27-s + 6·29-s − 4·33-s − 8·35-s − 10·37-s − 4·39-s − 6·41-s + 6·43-s − 2·45-s + 8·47-s + 9·49-s − 4·51-s + 6·53-s + 4·55-s + 4·57-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 1.03·15-s − 0.485·17-s + 0.458·19-s + 1.74·21-s − 0.834·23-s − 1/5·25-s − 0.769·27-s + 1.11·29-s − 0.696·33-s − 1.35·35-s − 1.64·37-s − 0.640·39-s − 0.937·41-s + 0.914·43-s − 0.298·45-s + 1.16·47-s + 9/7·49-s − 0.560·51-s + 0.824·53-s + 0.539·55-s + 0.529·57-s + ⋯

Functional equation

Λ(s)=(128s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(128s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 128128    =    272^{7}
Sign: 11
Analytic conductor: 1.022081.02208
Root analytic conductor: 1.010981.01098
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 128, ( :1/2), 1)(2,\ 128,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3736768691.373676869
L(12)L(\frac12) ≈\approx 1.3736768691.373676869
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
good3 1−2T+pT2 1 - 2 T + p T^{2}
5 1+2T+pT2 1 + 2 T + p T^{2}
7 1−4T+pT2 1 - 4 T + 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+2T+pT2 1 + 2 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−14T+pT2 1 - 14 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1+2T+pT2 1 + 2 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

−13.71655139960009945542339560653, −12.20334089315653888918962088761, −11.41599623750261821832363423665, −10.22941768705266383849925633731, −8.728992519223720851294830795286, −8.098681382024412731658064042949, −7.33738117792255002879146989083, −5.16542238567969836254694300670, −3.91058908136761919842529082491, −2.30676446591740616355718333113, 2.30676446591740616355718333113, 3.91058908136761919842529082491, 5.16542238567969836254694300670, 7.33738117792255002879146989083, 8.098681382024412731658064042949, 8.728992519223720851294830795286, 10.22941768705266383849925633731, 11.41599623750261821832363423665, 12.20334089315653888918962088761, 13.71655139960009945542339560653

Graph of the ZZ-function along the critical line