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Properties

Label 2-2156-1.1-c3-0-40
Degree 22
Conductor 21562156
Sign 11
Analytic cond. 127.208127.208
Root an. cond. 11.278611.2786
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·3-s + 7·5-s − 2·9-s − 11·11-s − 52·13-s + 35·15-s − 46·17-s + 96·19-s + 27·23-s − 76·25-s − 145·27-s + 16·29-s + 293·31-s − 55·33-s − 29·37-s − 260·39-s + 472·41-s − 110·43-s − 14·45-s + 224·47-s − 230·51-s + 754·53-s − 77·55-s + 480·57-s − 825·59-s + 548·61-s − 364·65-s + ⋯
L(s)  = 1  + 0.962·3-s + 0.626·5-s − 0.0740·9-s − 0.301·11-s − 1.10·13-s + 0.602·15-s − 0.656·17-s + 1.15·19-s + 0.244·23-s − 0.607·25-s − 1.03·27-s + 0.102·29-s + 1.69·31-s − 0.290·33-s − 0.128·37-s − 1.06·39-s + 1.79·41-s − 0.390·43-s − 0.0463·45-s + 0.695·47-s − 0.631·51-s + 1.95·53-s − 0.188·55-s + 1.11·57-s − 1.82·59-s + 1.15·61-s − 0.694·65-s + ⋯

Functional equation

Λ(s)=(2156s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2156s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21562156    =    22⋅72⋅112^{2} \cdot 7^{2} \cdot 11
Sign: 11
Analytic conductor: 127.208127.208
Root analytic conductor: 11.278611.2786
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2156, ( :3/2), 1)(2,\ 2156,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.2219208723.221920872
L(12)L(\frac12) ≈\approx 3.2219208723.221920872
L(52)L(\frac{5}{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
7 1 1
11 1+pT 1 + p T
good3 1−5T+p3T2 1 - 5 T + p^{3} T^{2}
5 1−7T+p3T2 1 - 7 T + p^{3} T^{2}
13 1+4pT+p3T2 1 + 4 p T + p^{3} T^{2}
17 1+46T+p3T2 1 + 46 T + p^{3} T^{2}
19 1−96T+p3T2 1 - 96 T + p^{3} T^{2}
23 1−27T+p3T2 1 - 27 T + p^{3} T^{2}
29 1−16T+p3T2 1 - 16 T + p^{3} T^{2}
31 1−293T+p3T2 1 - 293 T + p^{3} T^{2}
37 1+29T+p3T2 1 + 29 T + p^{3} T^{2}
41 1−472T+p3T2 1 - 472 T + p^{3} T^{2}
43 1+110T+p3T2 1 + 110 T + p^{3} T^{2}
47 1−224T+p3T2 1 - 224 T + p^{3} T^{2}
53 1−754T+p3T2 1 - 754 T + p^{3} T^{2}
59 1+825T+p3T2 1 + 825 T + p^{3} T^{2}
61 1−548T+p3T2 1 - 548 T + p^{3} T^{2}
67 1+123T+p3T2 1 + 123 T + p^{3} T^{2}
71 1−1001T+p3T2 1 - 1001 T + p^{3} T^{2}
73 1−1020T+p3T2 1 - 1020 T + p^{3} T^{2}
79 1−526T+p3T2 1 - 526 T + p^{3} T^{2}
83 1−158T+p3T2 1 - 158 T + p^{3} T^{2}
89 1−1217T+p3T2 1 - 1217 T + p^{3} T^{2}
97 1−263T+p3T2 1 - 263 T + p^{3} 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

−8.794266427745288570650119341614, −7.930730449002280849453570115533, −7.39899213381896901552690483251, −6.39562919451500966656812411301, −5.51238494649100799433347544718, −4.72825591134328527969285361270, −3.64173091144746530279267835125, −2.60230011025103489382875352299, −2.22735929181966407734513557795, −0.75493905879002333244016796085, 0.75493905879002333244016796085, 2.22735929181966407734513557795, 2.60230011025103489382875352299, 3.64173091144746530279267835125, 4.72825591134328527969285361270, 5.51238494649100799433347544718, 6.39562919451500966656812411301, 7.39899213381896901552690483251, 7.930730449002280849453570115533, 8.794266427745288570650119341614

Graph of the ZZ-function along the critical line