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Properties

Label 2-5760-1.1-c1-0-25
Degree 22
Conductor 57605760
Sign 11
Analytic cond. 45.993845.9938
Root an. cond. 6.781876.78187
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  + 5-s − 4·7-s + 6·11-s + 4·13-s − 4·17-s + 8·19-s + 25-s + 2·29-s − 2·31-s − 4·35-s − 4·37-s − 6·41-s + 12·43-s + 9·49-s − 14·53-s + 6·55-s − 6·59-s + 6·61-s + 4·65-s + 4·67-s + 8·71-s − 2·73-s − 24·77-s + 6·79-s + 12·83-s − 4·85-s − 6·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s + 1.80·11-s + 1.10·13-s − 0.970·17-s + 1.83·19-s + 1/5·25-s + 0.371·29-s − 0.359·31-s − 0.676·35-s − 0.657·37-s − 0.937·41-s + 1.82·43-s + 9/7·49-s − 1.92·53-s + 0.809·55-s − 0.781·59-s + 0.768·61-s + 0.496·65-s + 0.488·67-s + 0.949·71-s − 0.234·73-s − 2.73·77-s + 0.675·79-s + 1.31·83-s − 0.433·85-s − 0.635·89-s + ⋯

Functional equation

Λ(s)=(5760s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5760s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57605760    =    27⋅32⋅52^{7} \cdot 3^{2} \cdot 5
Sign: 11
Analytic conductor: 45.993845.9938
Root analytic conductor: 6.781876.78187
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5760, ( :1/2), 1)(2,\ 5760,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1718024052.171802405
L(12)L(\frac12) ≈\approx 2.1718024052.171802405
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
3 1 1
5 1−T 1 - T
good7 1+4T+pT2 1 + 4 T + p T^{2}
11 1−6T+pT2 1 - 6 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1−8T+pT2 1 - 8 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−12T+pT2 1 - 12 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+14T+pT2 1 + 14 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1−6T+pT2 1 - 6 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+14T+pT2 1 + 14 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

−8.233707745846371238456044860170, −7.09752660423814295441357875001, −6.63051434487844039149409502075, −6.16160510715634515135325957529, −5.42672318398502146703592000043, −4.28389757470006853731361576226, −3.55192887194675292853371851200, −3.04129716998461074149844302940, −1.74072122715861936107235590781, −0.818171558753717111286502894239, 0.818171558753717111286502894239, 1.74072122715861936107235590781, 3.04129716998461074149844302940, 3.55192887194675292853371851200, 4.28389757470006853731361576226, 5.42672318398502146703592000043, 6.16160510715634515135325957529, 6.63051434487844039149409502075, 7.09752660423814295441357875001, 8.233707745846371238456044860170

Graph of the ZZ-function along the critical line