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Properties

Label 2-2e8-1.1-c1-0-4
Degree 22
Conductor 256256
Sign −1-1
Analytic cond. 2.044172.04417
Root an. cond. 1.429741.42974
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 + 9-s − 6·11-s − 6·17-s − 2·19-s − 5·25-s + 4·27-s + 12·33-s + 6·41-s + 10·43-s − 7·49-s + 12·51-s + 4·57-s − 6·59-s + 14·67-s − 2·73-s + 10·75-s − 11·81-s − 18·83-s − 18·89-s + 10·97-s − 6·99-s − 6·107-s + 18·113-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 1.80·11-s − 1.45·17-s − 0.458·19-s − 25-s + 0.769·27-s + 2.08·33-s + 0.937·41-s + 1.52·43-s − 49-s + 1.68·51-s + 0.529·57-s − 0.781·59-s + 1.71·67-s − 0.234·73-s + 1.15·75-s − 1.22·81-s − 1.97·83-s − 1.90·89-s + 1.01·97-s − 0.603·99-s − 0.580·107-s + 1.69·113-s + ⋯

Functional equation

Λ(s)=(256s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(256s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 256256    =    282^{8}
Sign: −1-1
Analytic conductor: 2.044172.04417
Root analytic conductor: 1.429741.42974
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 256, ( :1/2), −1)(2,\ 256,\ (\ :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
good3 1+2T+pT2 1 + 2 T + p T^{2}
5 1+pT2 1 + p T^{2}
7 1+pT2 1 + p T^{2}
11 1+6T+pT2 1 + 6 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+pT2 1 + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−10T+pT2 1 - 10 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+pT2 1 + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1−14T+pT2 1 - 14 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+18T+pT2 1 + 18 T + p T^{2}
89 1+18T+pT2 1 + 18 T + p T^{2}
97 1−10T+pT2 1 - 10 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

−11.26097531340688560390760400117, −10.88783381933864566755233769368, −9.891997921642981576990976420061, −8.559961689771162774439116698021, −7.49638267192940041493128754308, −6.29051795613585298614624812000, −5.43412469058090981748265870176, −4.43148007247498493089277458199, −2.49619173311123854201331757855, 0, 2.49619173311123854201331757855, 4.43148007247498493089277458199, 5.43412469058090981748265870176, 6.29051795613585298614624812000, 7.49638267192940041493128754308, 8.559961689771162774439116698021, 9.891997921642981576990976420061, 10.88783381933864566755233769368, 11.26097531340688560390760400117

Graph of the ZZ-function along the critical line