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Properties

Label 2-2028-1.1-c1-0-9
Degree 22
Conductor 20282028
Sign 11
Analytic cond. 16.193616.1936
Root an. cond. 4.024134.02413
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 + 2·5-s − 4·7-s + 9-s + 6·11-s + 2·15-s + 2·17-s − 4·21-s + 8·23-s − 25-s + 27-s + 2·29-s − 8·31-s + 6·33-s − 8·35-s − 8·37-s − 2·41-s + 8·43-s + 2·45-s + 6·47-s + 9·49-s + 2·51-s + 6·53-s + 12·55-s + 2·59-s + 2·61-s − 4·63-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s + 1.80·11-s + 0.516·15-s + 0.485·17-s − 0.872·21-s + 1.66·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 1.04·33-s − 1.35·35-s − 1.31·37-s − 0.312·41-s + 1.21·43-s + 0.298·45-s + 0.875·47-s + 9/7·49-s + 0.280·51-s + 0.824·53-s + 1.61·55-s + 0.260·59-s + 0.256·61-s − 0.503·63-s + ⋯

Functional equation

Λ(s)=(2028s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2028s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20282028    =    22⋅3⋅1322^{2} \cdot 3 \cdot 13^{2}
Sign: 11
Analytic conductor: 16.193616.1936
Root analytic conductor: 4.024134.02413
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2028, ( :1/2), 1)(2,\ 2028,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5015925602.501592560
L(12)L(\frac12) ≈\approx 2.5015925602.501592560
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)Isogeny Class over Fp\mathbf{F}_p
bad2 1 1
3 1−T 1 - T
13 1 1
good5 1−2T+pT2 1 - 2 T + p T^{2} 1.5.ac
7 1+4T+pT2 1 + 4 T + p T^{2} 1.7.e
11 1−6T+pT2 1 - 6 T + p T^{2} 1.11.ag
17 1−2T+pT2 1 - 2 T + p T^{2} 1.17.ac
19 1+pT2 1 + p T^{2} 1.19.a
23 1−8T+pT2 1 - 8 T + p T^{2} 1.23.ai
29 1−2T+pT2 1 - 2 T + p T^{2} 1.29.ac
31 1+8T+pT2 1 + 8 T + p T^{2} 1.31.i
37 1+8T+pT2 1 + 8 T + p T^{2} 1.37.i
41 1+2T+pT2 1 + 2 T + p T^{2} 1.41.c
43 1−8T+pT2 1 - 8 T + p T^{2} 1.43.ai
47 1−6T+pT2 1 - 6 T + p T^{2} 1.47.ag
53 1−6T+pT2 1 - 6 T + p T^{2} 1.53.ag
59 1−2T+pT2 1 - 2 T + p T^{2} 1.59.ac
61 1−2T+pT2 1 - 2 T + p T^{2} 1.61.ac
67 1−4T+pT2 1 - 4 T + p T^{2} 1.67.ae
71 1−6T+pT2 1 - 6 T + p T^{2} 1.71.ag
73 1−4T+pT2 1 - 4 T + p T^{2} 1.73.ae
79 1+pT2 1 + p T^{2} 1.79.a
83 1−14T+pT2 1 - 14 T + p T^{2} 1.83.ao
89 1+6T+pT2 1 + 6 T + p T^{2} 1.89.g
97 1+12T+pT2 1 + 12 T + p T^{2} 1.97.m
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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

−9.097345964318492334723562115689, −8.847995063019436691526938067713, −7.34210278452480828621204466217, −6.77539691873077915389895911176, −6.13802705357204697531670904552, −5.26231588330551344370909293821, −3.87277603536411382354108264341, −3.37774050268429728726511902097, −2.27746946626923832361262697439, −1.09332650405258333005382089074, 1.09332650405258333005382089074, 2.27746946626923832361262697439, 3.37774050268429728726511902097, 3.87277603536411382354108264341, 5.26231588330551344370909293821, 6.13802705357204697531670904552, 6.77539691873077915389895911176, 7.34210278452480828621204466217, 8.847995063019436691526938067713, 9.097345964318492334723562115689

Graph of the ZZ-function along the critical line