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Properties

Label 2-4896-1.1-c1-0-53
Degree 22
Conductor 48964896
Sign −1-1
Analytic cond. 39.094739.0947
Root an. cond. 6.252586.25258
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  − 2·7-s − 6·13-s + 17-s + 4·19-s + 6·23-s − 5·25-s + 8·29-s + 2·31-s + 12·37-s + 2·41-s − 12·43-s − 4·47-s − 3·49-s + 14·53-s − 12·59-s − 4·61-s − 4·67-s − 6·71-s − 6·73-s − 14·79-s − 12·83-s − 6·89-s + 12·91-s − 10·97-s − 2·101-s + 8·103-s − 16·107-s + ⋯
L(s)  = 1  − 0.755·7-s − 1.66·13-s + 0.242·17-s + 0.917·19-s + 1.25·23-s − 25-s + 1.48·29-s + 0.359·31-s + 1.97·37-s + 0.312·41-s − 1.82·43-s − 0.583·47-s − 3/7·49-s + 1.92·53-s − 1.56·59-s − 0.512·61-s − 0.488·67-s − 0.712·71-s − 0.702·73-s − 1.57·79-s − 1.31·83-s − 0.635·89-s + 1.25·91-s − 1.01·97-s − 0.199·101-s + 0.788·103-s − 1.54·107-s + ⋯

Functional equation

Λ(s)=(4896s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4896s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 48964896    =    25⋅32⋅172^{5} \cdot 3^{2} \cdot 17
Sign: −1-1
Analytic conductor: 39.094739.0947
Root analytic conductor: 6.252586.25258
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4896, ( :1/2), −1)(2,\ 4896,\ (\ :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
17 1−T 1 - T
good5 1+pT2 1 + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1−8T+pT2 1 - 8 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1−12T+pT2 1 - 12 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+12T+pT2 1 + 12 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1−14T+pT2 1 - 14 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1+4T+pT2 1 + 4 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1+14T+pT2 1 + 14 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+10T+pT2 1 + 10 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

−7.77718952625556122797432773717, −7.20687508770729629161450298415, −6.53969561339537980266253033760, −5.71174706786893053338055753769, −4.92413858868962351564688633033, −4.26944062685429511445546705704, −3.01092071925155918358443492679, −2.72778690415535571819513199305, −1.29080808634123104616625472472, 0, 1.29080808634123104616625472472, 2.72778690415535571819513199305, 3.01092071925155918358443492679, 4.26944062685429511445546705704, 4.92413858868962351564688633033, 5.71174706786893053338055753769, 6.53969561339537980266253033760, 7.20687508770729629161450298415, 7.77718952625556122797432773717

Graph of the ZZ-function along the critical line