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Properties

Label 2-4788-1.1-c1-0-42
Degree 22
Conductor 47884788
Sign −1-1
Analytic cond. 38.232338.2323
Root an. cond. 6.183236.18323
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  + 2·5-s + 7-s − 4·11-s + 4·13-s − 6·17-s + 19-s − 4·23-s − 25-s − 6·29-s + 4·31-s + 2·35-s − 10·37-s − 4·41-s − 8·43-s + 49-s − 10·53-s − 8·55-s + 4·59-s + 14·61-s + 8·65-s − 6·67-s + 6·71-s − 2·73-s − 4·77-s − 10·79-s + 4·83-s − 12·85-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s − 1.20·11-s + 1.10·13-s − 1.45·17-s + 0.229·19-s − 0.834·23-s − 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.338·35-s − 1.64·37-s − 0.624·41-s − 1.21·43-s + 1/7·49-s − 1.37·53-s − 1.07·55-s + 0.520·59-s + 1.79·61-s + 0.992·65-s − 0.733·67-s + 0.712·71-s − 0.234·73-s − 0.455·77-s − 1.12·79-s + 0.439·83-s − 1.30·85-s + ⋯

Functional equation

Λ(s)=(4788s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4788s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 47884788    =    22⋅32⋅7⋅192^{2} \cdot 3^{2} \cdot 7 \cdot 19
Sign: −1-1
Analytic conductor: 38.232338.2323
Root analytic conductor: 6.183236.18323
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4788, ( :1/2), −1)(2,\ 4788,\ (\ :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 1
7 1−T 1 - T
19 1−T 1 - T
good5 1−2T+pT2 1 - 2 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1+4T+pT2 1 + 4 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1+6T+pT2 1 + 6 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+12T+pT2 1 + 12 T + p T^{2}
97 1+12T+pT2 1 + 12 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

−8.167412292899053112083531589323, −7.10637580445365728452881610550, −6.43908761746468281585509954459, −5.66190834171095963067447872745, −5.14275368814923235962222968050, −4.21195947144523985076795578238, −3.27818049362169095485162406589, −2.20595699566621536787083813145, −1.64118230998341906462975859295, 0, 1.64118230998341906462975859295, 2.20595699566621536787083813145, 3.27818049362169095485162406589, 4.21195947144523985076795578238, 5.14275368814923235962222968050, 5.66190834171095963067447872745, 6.43908761746468281585509954459, 7.10637580445365728452881610550, 8.167412292899053112083531589323

Graph of the ZZ-function along the critical line