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Properties

Label 2-504-1.1-c5-0-37
Degree 22
Conductor 504504
Sign −1-1
Analytic cond. 80.833480.8334
Root an. cond. 8.990748.99074
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 81.4·5-s + 49·7-s + 340.·11-s − 1.10e3·13-s + 197.·17-s − 2.33e3·19-s − 2.60e3·23-s + 3.50e3·25-s − 7.91e3·29-s − 9.04e3·31-s + 3.98e3·35-s − 5.47e3·37-s + 1.52e4·41-s − 3.82e3·43-s − 1.94e3·47-s + 2.40e3·49-s − 2.62e4·53-s + 2.77e4·55-s + 4.58e4·59-s − 4.34e4·61-s − 8.96e4·65-s + 1.58e4·67-s + 2.41e4·71-s + 6.90e4·73-s + 1.66e4·77-s + 6.09e4·79-s − 5.22e4·83-s + ⋯
L(s)  = 1  + 1.45·5-s + 0.377·7-s + 0.847·11-s − 1.80·13-s + 0.165·17-s − 1.48·19-s − 1.02·23-s + 1.12·25-s − 1.74·29-s − 1.69·31-s + 0.550·35-s − 0.657·37-s + 1.41·41-s − 0.315·43-s − 0.128·47-s + 0.142·49-s − 1.28·53-s + 1.23·55-s + 1.71·59-s − 1.49·61-s − 2.63·65-s + 0.431·67-s + 0.567·71-s + 1.51·73-s + 0.320·77-s + 1.09·79-s − 0.832·83-s + ⋯

Functional equation

Λ(s)=(504s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(504s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 504504    =    23⋅32⋅72^{3} \cdot 3^{2} \cdot 7
Sign: −1-1
Analytic conductor: 80.833480.8334
Root analytic conductor: 8.990748.99074
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 504, ( :5/2), −1)(2,\ 504,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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
7 1−49T 1 - 49T
good5 1−81.4T+3.12e3T2 1 - 81.4T + 3.12e3T^{2}
11 1−340.T+1.61e5T2 1 - 340.T + 1.61e5T^{2}
13 1+1.10e3T+3.71e5T2 1 + 1.10e3T + 3.71e5T^{2}
17 1−197.T+1.41e6T2 1 - 197.T + 1.41e6T^{2}
19 1+2.33e3T+2.47e6T2 1 + 2.33e3T + 2.47e6T^{2}
23 1+2.60e3T+6.43e6T2 1 + 2.60e3T + 6.43e6T^{2}
29 1+7.91e3T+2.05e7T2 1 + 7.91e3T + 2.05e7T^{2}
31 1+9.04e3T+2.86e7T2 1 + 9.04e3T + 2.86e7T^{2}
37 1+5.47e3T+6.93e7T2 1 + 5.47e3T + 6.93e7T^{2}
41 1−1.52e4T+1.15e8T2 1 - 1.52e4T + 1.15e8T^{2}
43 1+3.82e3T+1.47e8T2 1 + 3.82e3T + 1.47e8T^{2}
47 1+1.94e3T+2.29e8T2 1 + 1.94e3T + 2.29e8T^{2}
53 1+2.62e4T+4.18e8T2 1 + 2.62e4T + 4.18e8T^{2}
59 1−4.58e4T+7.14e8T2 1 - 4.58e4T + 7.14e8T^{2}
61 1+4.34e4T+8.44e8T2 1 + 4.34e4T + 8.44e8T^{2}
67 1−1.58e4T+1.35e9T2 1 - 1.58e4T + 1.35e9T^{2}
71 1−2.41e4T+1.80e9T2 1 - 2.41e4T + 1.80e9T^{2}
73 1−6.90e4T+2.07e9T2 1 - 6.90e4T + 2.07e9T^{2}
79 1−6.09e4T+3.07e9T2 1 - 6.09e4T + 3.07e9T^{2}
83 1+5.22e4T+3.93e9T2 1 + 5.22e4T + 3.93e9T^{2}
89 1+1.00e5T+5.58e9T2 1 + 1.00e5T + 5.58e9T^{2}
97 1−6.49e4T+8.58e9T2 1 - 6.49e4T + 8.58e9T^{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

−9.567767178512870587682137245733, −9.131036245862864957765690512170, −7.81985531457090474611893169462, −6.83922864900154506249809388919, −5.90034977237662648862442148715, −5.09822234484879880557327616876, −3.93240924243128955304812489077, −2.26108204243016073910759863426, −1.77469360753255666603762645130, 0, 1.77469360753255666603762645130, 2.26108204243016073910759863426, 3.93240924243128955304812489077, 5.09822234484879880557327616876, 5.90034977237662648862442148715, 6.83922864900154506249809388919, 7.81985531457090474611893169462, 9.131036245862864957765690512170, 9.567767178512870587682137245733

Graph of the ZZ-function along the critical line