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Properties

Label 2-1040-1.1-c5-0-101
Degree 22
Conductor 10401040
Sign −1-1
Analytic cond. 166.799166.799
Root an. cond. 12.915012.9150
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  + 19.6·3-s − 25·5-s − 48.6·7-s + 142.·9-s − 283.·11-s + 169·13-s − 491.·15-s + 789.·17-s − 83.3·19-s − 955.·21-s + 1.93e3·23-s + 625·25-s − 1.96e3·27-s + 222.·29-s + 2.78e3·31-s − 5.56e3·33-s + 1.21e3·35-s + 8.36e3·37-s + 3.31e3·39-s + 1.21e3·41-s − 5.63e3·43-s − 3.57e3·45-s − 1.77e4·47-s − 1.44e4·49-s + 1.55e4·51-s + 1.08e4·53-s + 7.08e3·55-s + ⋯
L(s)  = 1  + 1.26·3-s − 0.447·5-s − 0.375·7-s + 0.587·9-s − 0.706·11-s + 0.277·13-s − 0.563·15-s + 0.662·17-s − 0.0529·19-s − 0.472·21-s + 0.762·23-s + 0.200·25-s − 0.519·27-s + 0.0491·29-s + 0.521·31-s − 0.890·33-s + 0.167·35-s + 1.00·37-s + 0.349·39-s + 0.113·41-s − 0.465·43-s − 0.262·45-s − 1.17·47-s − 0.859·49-s + 0.834·51-s + 0.529·53-s + 0.315·55-s + ⋯

Functional equation

Λ(s)=(1040s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(1040s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10401040    =    24⋅5⋅132^{4} \cdot 5 \cdot 13
Sign: −1-1
Analytic conductor: 166.799166.799
Root analytic conductor: 12.915012.9150
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1040, ( :5/2), −1)(2,\ 1040,\ (\ :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
5 1+25T 1 + 25T
13 1−169T 1 - 169T
good3 1−19.6T+243T2 1 - 19.6T + 243T^{2}
7 1+48.6T+1.68e4T2 1 + 48.6T + 1.68e4T^{2}
11 1+283.T+1.61e5T2 1 + 283.T + 1.61e5T^{2}
17 1−789.T+1.41e6T2 1 - 789.T + 1.41e6T^{2}
19 1+83.3T+2.47e6T2 1 + 83.3T + 2.47e6T^{2}
23 1−1.93e3T+6.43e6T2 1 - 1.93e3T + 6.43e6T^{2}
29 1−222.T+2.05e7T2 1 - 222.T + 2.05e7T^{2}
31 1−2.78e3T+2.86e7T2 1 - 2.78e3T + 2.86e7T^{2}
37 1−8.36e3T+6.93e7T2 1 - 8.36e3T + 6.93e7T^{2}
41 1−1.21e3T+1.15e8T2 1 - 1.21e3T + 1.15e8T^{2}
43 1+5.63e3T+1.47e8T2 1 + 5.63e3T + 1.47e8T^{2}
47 1+1.77e4T+2.29e8T2 1 + 1.77e4T + 2.29e8T^{2}
53 1−1.08e4T+4.18e8T2 1 - 1.08e4T + 4.18e8T^{2}
59 1−5.36e3T+7.14e8T2 1 - 5.36e3T + 7.14e8T^{2}
61 1+1.86e4T+8.44e8T2 1 + 1.86e4T + 8.44e8T^{2}
67 1+1.39e4T+1.35e9T2 1 + 1.39e4T + 1.35e9T^{2}
71 1+5.09e4T+1.80e9T2 1 + 5.09e4T + 1.80e9T^{2}
73 1−4.23e4T+2.07e9T2 1 - 4.23e4T + 2.07e9T^{2}
79 1+1.06e5T+3.07e9T2 1 + 1.06e5T + 3.07e9T^{2}
83 1+7.55e4T+3.93e9T2 1 + 7.55e4T + 3.93e9T^{2}
89 1−7.70e4T+5.58e9T2 1 - 7.70e4T + 5.58e9T^{2}
97 1−1.26e5T+8.58e9T2 1 - 1.26e5T + 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

−8.667113304752254855741038512910, −8.031015963332111296650441102436, −7.40707137502712305678876286074, −6.35750892470555127557458349317, −5.23887600500557711393328713686, −4.15075731845071741741851325060, −3.18633688900167520149547058053, −2.67721104588665522663585575467, −1.35510861283643087878898918155, 0, 1.35510861283643087878898918155, 2.67721104588665522663585575467, 3.18633688900167520149547058053, 4.15075731845071741741851325060, 5.23887600500557711393328713686, 6.35750892470555127557458349317, 7.40707137502712305678876286074, 8.031015963332111296650441102436, 8.667113304752254855741038512910

Graph of the ZZ-function along the critical line