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Properties

Label 2-2496-1.1-c1-0-35
Degree 22
Conductor 24962496
Sign −1-1
Analytic cond. 19.930619.9306
Root an. cond. 4.464374.46437
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·7-s + 9-s − 13-s − 6·17-s − 2·19-s − 2·21-s − 5·25-s − 27-s + 6·29-s + 2·31-s − 2·37-s + 39-s − 12·41-s + 4·43-s − 3·49-s + 6·51-s − 6·53-s + 2·57-s − 12·59-s − 2·61-s + 2·63-s + 10·67-s + 12·71-s + 14·73-s + 5·75-s + 8·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.277·13-s − 1.45·17-s − 0.458·19-s − 0.436·21-s − 25-s − 0.192·27-s + 1.11·29-s + 0.359·31-s − 0.328·37-s + 0.160·39-s − 1.87·41-s + 0.609·43-s − 3/7·49-s + 0.840·51-s − 0.824·53-s + 0.264·57-s − 1.56·59-s − 0.256·61-s + 0.251·63-s + 1.22·67-s + 1.42·71-s + 1.63·73-s + 0.577·75-s + 0.900·79-s + ⋯

Functional equation

Λ(s)=(2496s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2496s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24962496    =    26⋅3⋅132^{6} \cdot 3 \cdot 13
Sign: −1-1
Analytic conductor: 19.930619.9306
Root analytic conductor: 4.464374.46437
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2496, ( :1/2), −1)(2,\ 2496,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
13 1+T 1 + T
good5 1+pT2 1 + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+12T+pT2 1 + 12 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1−12T+pT2 1 - 12 T + 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+pT2 1 + 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

−8.397749324163862668074879881731, −7.903021947901095999330816312888, −6.77588051345062669609369199457, −6.38841228560375566367348698112, −5.23796721936294605278821689684, −4.69511531925842197635869476162, −3.86363777727094941125777069865, −2.49852565048742461389380956784, −1.54822382936130812607770538657, 0, 1.54822382936130812607770538657, 2.49852565048742461389380956784, 3.86363777727094941125777069865, 4.69511531925842197635869476162, 5.23796721936294605278821689684, 6.38841228560375566367348698112, 6.77588051345062669609369199457, 7.903021947901095999330816312888, 8.397749324163862668074879881731

Graph of the ZZ-function along the critical line