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Properties

Label 2-29760-1.1-c1-0-28
Degree 22
Conductor 2976029760
Sign 11
Analytic cond. 237.634237.634
Root an. cond. 15.415415.4154
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  + 3-s − 5-s + 9-s + 2·13-s − 15-s + 6·17-s + 4·19-s − 4·23-s + 25-s + 27-s − 2·29-s + 31-s + 2·37-s + 2·39-s − 6·41-s − 4·43-s − 45-s − 7·49-s + 6·51-s + 10·53-s + 4·57-s + 12·59-s − 10·61-s − 2·65-s + 4·67-s − 4·69-s + 14·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 1.45·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.179·31-s + 0.328·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s − 49-s + 0.840·51-s + 1.37·53-s + 0.529·57-s + 1.56·59-s − 1.28·61-s − 0.248·65-s + 0.488·67-s − 0.481·69-s + 1.63·73-s + ⋯

Functional equation

Λ(s)=(29760s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 29760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(29760s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 29760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2976029760    =    26⋅3⋅5⋅312^{6} \cdot 3 \cdot 5 \cdot 31
Sign: 11
Analytic conductor: 237.634237.634
Root analytic conductor: 15.415415.4154
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 29760, ( :1/2), 1)(2,\ 29760,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.9917737662.991773766
L(12)L(\frac12) ≈\approx 2.9917737662.991773766
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−T 1 - T
5 1+T 1 + T
31 1−T 1 - T
good7 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−10T+pT2 1 - 10 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

−15.01253576399936, −14.68934302541223, −14.11131747659472, −13.54421732564357, −13.21394556435156, −12.39687059239036, −11.90050079747187, −11.64611003080993, −10.71500930236369, −10.34568710200721, −9.537165270596482, −9.395860425017619, −8.371638156938433, −8.088602778787462, −7.636376930622977, −6.908151603633745, −6.348722181533343, −5.481563492473888, −5.117656210614012, −4.153399583000393, −3.567068723467665, −3.211997287139536, −2.286790944996329, −1.454759544044686, −0.6787475454761288, 0.6787475454761288, 1.454759544044686, 2.286790944996329, 3.211997287139536, 3.567068723467665, 4.153399583000393, 5.117656210614012, 5.481563492473888, 6.348722181533343, 6.908151603633745, 7.636376930622977, 8.088602778787462, 8.371638156938433, 9.395860425017619, 9.537165270596482, 10.34568710200721, 10.71500930236369, 11.64611003080993, 11.90050079747187, 12.39687059239036, 13.21394556435156, 13.54421732564357, 14.11131747659472, 14.68934302541223, 15.01253576399936

Graph of the ZZ-function along the critical line