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Properties

Label 2-29988-1.1-c1-0-39
Degree 22
Conductor 2998829988
Sign −1-1
Analytic cond. 239.455239.455
Root an. cond. 15.474315.4743
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 + 6·11-s + 13-s + 17-s − 5·19-s + 6·23-s − 25-s − 6·29-s − 3·31-s − 3·37-s − 9·43-s − 4·47-s − 2·53-s + 12·55-s + 4·59-s + 2·61-s + 2·65-s − 15·67-s − 13·73-s − 5·79-s + 2·85-s + 16·89-s − 10·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.80·11-s + 0.277·13-s + 0.242·17-s − 1.14·19-s + 1.25·23-s − 1/5·25-s − 1.11·29-s − 0.538·31-s − 0.493·37-s − 1.37·43-s − 0.583·47-s − 0.274·53-s + 1.61·55-s + 0.520·59-s + 0.256·61-s + 0.248·65-s − 1.83·67-s − 1.52·73-s − 0.562·79-s + 0.216·85-s + 1.69·89-s − 1.02·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

Λ(s)=(29988s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(29988s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2998829988    =    22⋅32⋅72⋅172^{2} \cdot 3^{2} \cdot 7^{2} \cdot 17
Sign: −1-1
Analytic conductor: 239.455239.455
Root analytic conductor: 15.474315.4743
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 29988, ( :1/2), −1)(2,\ 29988,\ (\ :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 1
17 1−T 1 - T
good5 1−2T+pT2 1 - 2 T + p T^{2}
11 1−6T+pT2 1 - 6 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
19 1+5T+pT2 1 + 5 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1+3T+pT2 1 + 3 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+9T+pT2 1 + 9 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1+15T+pT2 1 + 15 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+13T+pT2 1 + 13 T + p T^{2}
79 1+5T+pT2 1 + 5 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−16T+pT2 1 - 16 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

−15.03526318810891, −14.84027302145962, −14.53535931938352, −13.69554601437619, −13.34704940191964, −12.89674672411006, −12.15138345101089, −11.70959639720481, −11.10182100432892, −10.61350783600351, −9.917882066746630, −9.416653837193075, −8.939266993555096, −8.570494091092745, −7.693802051099673, −6.908230007470613, −6.593267338318689, −5.976225771015414, −5.433004342887227, −4.674911386760994, −3.947320715273473, −3.458418304141932, −2.562644883106160, −1.580360514822766, −1.428173114530598, 0, 1.428173114530598, 1.580360514822766, 2.562644883106160, 3.458418304141932, 3.947320715273473, 4.674911386760994, 5.433004342887227, 5.976225771015414, 6.593267338318689, 6.908230007470613, 7.693802051099673, 8.570494091092745, 8.939266993555096, 9.416653837193075, 9.917882066746630, 10.61350783600351, 11.10182100432892, 11.70959639720481, 12.15138345101089, 12.89674672411006, 13.34704940191964, 13.69554601437619, 14.53535931938352, 14.84027302145962, 15.03526318810891

Graph of the ZZ-function along the critical line