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Properties

Label 2-2e7-8.3-c0-0-0
Degree 22
Conductor 128128
Sign 11
Analytic cond. 0.06388030.0638803
Root an. cond. 0.2527450.252745
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9-s − 2·17-s + 25-s + 2·41-s + 49-s − 2·73-s + 81-s − 2·89-s − 2·97-s + 2·113-s + ⋯
L(s)  = 1  − 9-s − 2·17-s + 25-s + 2·41-s + 49-s − 2·73-s + 81-s − 2·89-s − 2·97-s + 2·113-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 128128    =    272^{7}
Sign: 11
Analytic conductor: 0.06388030.0638803
Root analytic conductor: 0.2527450.252745
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ128(63,⋅)\chi_{128} (63, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 128, ( :0), 1)(2,\ 128,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.58279163240.5827916324
L(12)L(\frac12) ≈\approx 0.58279163240.5827916324
L(1)L(1) 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+T2 1 + T^{2}
5 (1−T)(1+T) ( 1 - T )( 1 + T )
7 (1−T)(1+T) ( 1 - T )( 1 + T )
11 1+T2 1 + T^{2}
13 (1−T)(1+T) ( 1 - T )( 1 + T )
17 (1+T)2 ( 1 + T )^{2}
19 1+T2 1 + T^{2}
23 (1−T)(1+T) ( 1 - T )( 1 + T )
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 (1−T)(1+T) ( 1 - T )( 1 + T )
37 (1−T)(1+T) ( 1 - T )( 1 + T )
41 (1−T)2 ( 1 - T )^{2}
43 1+T2 1 + T^{2}
47 (1−T)(1+T) ( 1 - T )( 1 + T )
53 (1−T)(1+T) ( 1 - T )( 1 + T )
59 1+T2 1 + T^{2}
61 (1−T)(1+T) ( 1 - T )( 1 + T )
67 1+T2 1 + T^{2}
71 (1−T)(1+T) ( 1 - T )( 1 + T )
73 (1+T)2 ( 1 + T )^{2}
79 (1−T)(1+T) ( 1 - T )( 1 + T )
83 1+T2 1 + T^{2}
89 (1+T)2 ( 1 + T )^{2}
97 (1+T)2 ( 1 + 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

−13.60653957404905351544532534862, −12.62730928344066147828554111521, −11.40941624187610021847514464408, −10.72859342897654553267618939149, −9.220994759867439580089527677520, −8.461795530674634695923023014770, −7.02072783744560160256463601653, −5.84282968988336850818779137015, −4.41060008842499151557844234715, −2.63592829534740587398498486196, 2.63592829534740587398498486196, 4.41060008842499151557844234715, 5.84282968988336850818779137015, 7.02072783744560160256463601653, 8.461795530674634695923023014770, 9.220994759867439580089527677520, 10.72859342897654553267618939149, 11.40941624187610021847514464408, 12.62730928344066147828554111521, 13.60653957404905351544532534862

Graph of the ZZ-function along the critical line