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Properties

Label 2-1872-1.1-c1-0-10
Degree 22
Conductor 18721872
Sign 11
Analytic cond. 14.947914.9479
Root an. cond. 3.866263.86626
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  + 4·7-s − 2·11-s − 13-s + 6·17-s + 4·19-s + 4·23-s − 5·25-s − 10·29-s + 8·31-s − 2·37-s + 4·43-s + 2·47-s + 9·49-s + 2·53-s + 10·59-s + 10·61-s − 8·67-s + 2·71-s − 10·73-s − 8·77-s − 8·79-s + 6·83-s + 12·89-s − 4·91-s − 2·97-s − 2·101-s + 16·103-s + ⋯
L(s)  = 1  + 1.51·7-s − 0.603·11-s − 0.277·13-s + 1.45·17-s + 0.917·19-s + 0.834·23-s − 25-s − 1.85·29-s + 1.43·31-s − 0.328·37-s + 0.609·43-s + 0.291·47-s + 9/7·49-s + 0.274·53-s + 1.30·59-s + 1.28·61-s − 0.977·67-s + 0.237·71-s − 1.17·73-s − 0.911·77-s − 0.900·79-s + 0.658·83-s + 1.27·89-s − 0.419·91-s − 0.203·97-s − 0.199·101-s + 1.57·103-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: 11
Analytic conductor: 14.947914.9479
Root analytic conductor: 3.866263.86626
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1872, ( :1/2), 1)(2,\ 1872,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1303504952.130350495
L(12)L(\frac12) ≈\approx 2.1303504952.130350495
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 1
13 1+T 1 + T
good5 1+pT2 1 + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+10T+pT2 1 + 10 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−2T+pT2 1 - 2 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1−2T+pT2 1 - 2 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1+2T+pT2 1 + 2 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

−9.239289551943017123934612848385, −8.246518608998310493379441373007, −7.71387236722259124950052322681, −7.17292965380990182887285956413, −5.66602342897317417961265171945, −5.33322403312144877575207388138, −4.39131354020728293864682849714, −3.31414609411084864193635820227, −2.15956775014138729773769000571, −1.06254916932353976871281040816, 1.06254916932353976871281040816, 2.15956775014138729773769000571, 3.31414609411084864193635820227, 4.39131354020728293864682849714, 5.33322403312144877575207388138, 5.66602342897317417961265171945, 7.17292965380990182887285956413, 7.71387236722259124950052322681, 8.246518608998310493379441373007, 9.239289551943017123934612848385

Graph of the ZZ-function along the critical line