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Properties

Label 2-560-35.34-c2-0-28
Degree 22
Conductor 560560
Sign 11
Analytic cond. 15.258815.2588
Root an. cond. 3.906263.90626
Motivic weight 22
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5·5-s + 7·7-s − 8·9-s + 13·11-s + 19·13-s + 5·15-s − 29·17-s + 7·21-s + 25·25-s − 17·27-s + 23·29-s + 13·33-s + 35·35-s + 19·39-s − 40·45-s − 31·47-s + 49·49-s − 29·51-s + 65·55-s − 56·63-s + 95·65-s − 2·71-s + 34·73-s + 25·75-s + 91·77-s + 157·79-s + ⋯
L(s)  = 1  + 1/3·3-s + 5-s + 7-s − 8/9·9-s + 1.18·11-s + 1.46·13-s + 1/3·15-s − 1.70·17-s + 1/3·21-s + 25-s − 0.629·27-s + 0.793·29-s + 0.393·33-s + 35-s + 0.487·39-s − 8/9·45-s − 0.659·47-s + 49-s − 0.568·51-s + 1.18·55-s − 8/9·63-s + 1.46·65-s − 0.0281·71-s + 0.465·73-s + 1/3·75-s + 1.18·77-s + 1.98·79-s + ⋯

Functional equation

Λ(s)=(560s/2 ΓC(s) L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(560s/2 ΓC(s+1) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 560560    =    24⋅5⋅72^{4} \cdot 5 \cdot 7
Sign: 11
Analytic conductor: 15.258815.2588
Root analytic conductor: 3.906263.90626
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: χ560(209,⋅)\chi_{560} (209, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 560, ( :1), 1)(2,\ 560,\ (\ :1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.7226819662.722681966
L(12)L(\frac12) ≈\approx 2.7226819662.722681966
L(2)L(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
5 1−pT 1 - p T
7 1−pT 1 - p T
good3 1−T+p2T2 1 - T + p^{2} T^{2}
11 1−13T+p2T2 1 - 13 T + p^{2} T^{2}
13 1−19T+p2T2 1 - 19 T + p^{2} T^{2}
17 1+29T+p2T2 1 + 29 T + p^{2} T^{2}
19 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
23 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
29 1−23T+p2T2 1 - 23 T + p^{2} T^{2}
31 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
37 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
41 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
43 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
47 1+31T+p2T2 1 + 31 T + p^{2} T^{2}
53 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
59 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
61 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
67 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
71 1+2T+p2T2 1 + 2 T + p^{2} T^{2}
73 1−34T+p2T2 1 - 34 T + p^{2} T^{2}
79 1−157T+p2T2 1 - 157 T + p^{2} T^{2}
83 1−86T+p2T2 1 - 86 T + p^{2} T^{2}
89 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
97 1+149T+p2T2 1 + 149 T + p^{2} 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

−10.81774139496893165209540094579, −9.419704847457727036852785059137, −8.780413284014397940452448969180, −8.247705902108198010203357563341, −6.68960528705947591122207616241, −6.10040190795776390739412804847, −4.96712730622036284748748766932, −3.82847740500390756064826557597, −2.39859128913470652638872784326, −1.33088168508529191997763161761, 1.33088168508529191997763161761, 2.39859128913470652638872784326, 3.82847740500390756064826557597, 4.96712730622036284748748766932, 6.10040190795776390739412804847, 6.68960528705947591122207616241, 8.247705902108198010203357563341, 8.780413284014397940452448969180, 9.419704847457727036852785059137, 10.81774139496893165209540094579

Graph of the ZZ-function along the critical line