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Properties

Label 2-244800-1.1-c1-0-225
Degree 22
Conductor 244800244800
Sign −1-1
Analytic cond. 1954.731954.73
Root an. cond. 44.212444.2124
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  − 4·11-s − 2·13-s + 17-s − 4·19-s − 10·29-s + 8·31-s − 2·37-s − 10·41-s + 12·43-s − 7·49-s − 6·53-s + 12·59-s + 10·61-s − 12·67-s − 10·73-s − 8·79-s − 4·83-s + 6·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.20·11-s − 0.554·13-s + 0.242·17-s − 0.917·19-s − 1.85·29-s + 1.43·31-s − 0.328·37-s − 1.56·41-s + 1.82·43-s − 49-s − 0.824·53-s + 1.56·59-s + 1.28·61-s − 1.46·67-s − 1.17·73-s − 0.900·79-s − 0.439·83-s + 0.635·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

Λ(s)=(244800s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(244800s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 244800244800    =    26⋅32⋅52⋅172^{6} \cdot 3^{2} \cdot 5^{2} \cdot 17
Sign: −1-1
Analytic conductor: 1954.731954.73
Root analytic conductor: 44.212444.2124
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 244800, ( :1/2), −1)(2,\ 244800,\ (\ :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
5 1 1
17 1−T 1 - T
good7 1+pT2 1 + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+pT2 1 + 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+10T+pT2 1 + 10 T + p T^{2}
43 1−12T+pT2 1 - 12 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1+pT2 1 + 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+4T+pT2 1 + 4 T + p T^{2}
89 1−6T+pT2 1 - 6 T + 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

−12.99566229688687, −12.88496339987523, −12.15443229088822, −11.70094742933712, −11.31574222467888, −10.62585427050169, −10.43488219974738, −9.819761208656495, −9.566878901725527, −8.701257003365823, −8.528294504462651, −7.900295856794826, −7.412926507217181, −7.130292421533244, −6.365487615492384, −5.950154905061894, −5.380872861696427, −4.957342709797696, −4.423127136170162, −3.869508308322296, −3.189775343389660, −2.697040256840693, −2.120297236742076, −1.613583590369521, −0.6225288146000435, 0, 0.6225288146000435, 1.613583590369521, 2.120297236742076, 2.697040256840693, 3.189775343389660, 3.869508308322296, 4.423127136170162, 4.957342709797696, 5.380872861696427, 5.950154905061894, 6.365487615492384, 7.130292421533244, 7.412926507217181, 7.900295856794826, 8.528294504462651, 8.701257003365823, 9.566878901725527, 9.819761208656495, 10.43488219974738, 10.62585427050169, 11.31574222467888, 11.70094742933712, 12.15443229088822, 12.88496339987523, 12.99566229688687

Graph of the ZZ-function along the critical line