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Properties

Label 2-21e2-1.1-c1-0-3
Degree 22
Conductor 441441
Sign 11
Analytic cond. 3.521403.52140
Root an. cond. 1.876541.87654
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  − 2·4-s + 7·13-s + 4·16-s + 7·19-s − 5·25-s + 7·31-s − 37-s + 5·43-s − 14·52-s − 14·61-s − 8·64-s + 11·67-s + 7·73-s − 14·76-s − 13·79-s − 14·97-s + 10·100-s + 7·103-s + 17·109-s + ⋯
L(s)  = 1  − 4-s + 1.94·13-s + 16-s + 1.60·19-s − 25-s + 1.25·31-s − 0.164·37-s + 0.762·43-s − 1.94·52-s − 1.79·61-s − 64-s + 1.34·67-s + 0.819·73-s − 1.60·76-s − 1.46·79-s − 1.42·97-s + 100-s + 0.689·103-s + 1.62·109-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 3.521403.52140
Root analytic conductor: 1.876541.87654
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 441, ( :1/2), 1)(2,\ 441,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2091773641.209177364
L(12)L(\frac12) ≈\approx 1.2091773641.209177364
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)
bad3 1 1
7 1 1
good2 1+pT2 1 + p T^{2}
5 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−7T+pT2 1 - 7 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−7T+pT2 1 - 7 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−7T+pT2 1 - 7 T + p T^{2}
37 1+T+pT2 1 + T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−5T+pT2 1 - 5 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+pT2 1 + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+14T+pT2 1 + 14 T + p T^{2}
67 1−11T+pT2 1 - 11 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−7T+pT2 1 - 7 T + p T^{2}
79 1+13T+pT2 1 + 13 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+14T+pT2 1 + 14 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

−11.12611090306175738214534717990, −10.09209954751813487013591320496, −9.291988044548486732019479222726, −8.461373500657640359204806994178, −7.67458414128501773545216406647, −6.24141795466275696941914757951, −5.40791154265456502263047742620, −4.19234808180063134723072414533, −3.28050491254919886424015350098, −1.14668647924786773157165026874, 1.14668647924786773157165026874, 3.28050491254919886424015350098, 4.19234808180063134723072414533, 5.40791154265456502263047742620, 6.24141795466275696941914757951, 7.67458414128501773545216406647, 8.461373500657640359204806994178, 9.291988044548486732019479222726, 10.09209954751813487013591320496, 11.12611090306175738214534717990

Graph of the ZZ-function along the critical line