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Properties

Label 2-448-1.1-c3-0-28
Degree 22
Conductor 448448
Sign −1-1
Analytic cond. 26.432826.4328
Root an. cond. 5.141285.14128
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 6·5-s − 7·7-s − 11·9-s − 12·11-s + 82·13-s − 24·15-s − 30·17-s + 68·19-s − 28·21-s − 216·23-s − 89·25-s − 152·27-s − 246·29-s + 112·31-s − 48·33-s + 42·35-s − 110·37-s + 328·39-s − 246·41-s − 172·43-s + 66·45-s − 192·47-s + 49·49-s − 120·51-s − 558·53-s + 72·55-s + ⋯
L(s)  = 1  + 0.769·3-s − 0.536·5-s − 0.377·7-s − 0.407·9-s − 0.328·11-s + 1.74·13-s − 0.413·15-s − 0.428·17-s + 0.821·19-s − 0.290·21-s − 1.95·23-s − 0.711·25-s − 1.08·27-s − 1.57·29-s + 0.648·31-s − 0.253·33-s + 0.202·35-s − 0.488·37-s + 1.34·39-s − 0.937·41-s − 0.609·43-s + 0.218·45-s − 0.595·47-s + 1/7·49-s − 0.329·51-s − 1.44·53-s + 0.176·55-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: −1-1
Analytic conductor: 26.432826.4328
Root analytic conductor: 5.141285.14128
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 448, ( :3/2), −1)(2,\ 448,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
7 1+pT 1 + p T
good3 1−4T+p3T2 1 - 4 T + p^{3} T^{2}
5 1+6T+p3T2 1 + 6 T + p^{3} T^{2}
11 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
13 1−82T+p3T2 1 - 82 T + p^{3} T^{2}
17 1+30T+p3T2 1 + 30 T + p^{3} T^{2}
19 1−68T+p3T2 1 - 68 T + p^{3} T^{2}
23 1+216T+p3T2 1 + 216 T + p^{3} T^{2}
29 1+246T+p3T2 1 + 246 T + p^{3} T^{2}
31 1−112T+p3T2 1 - 112 T + p^{3} T^{2}
37 1+110T+p3T2 1 + 110 T + p^{3} T^{2}
41 1+6pT+p3T2 1 + 6 p T + p^{3} T^{2}
43 1+4pT+p3T2 1 + 4 p T + p^{3} T^{2}
47 1+192T+p3T2 1 + 192 T + p^{3} T^{2}
53 1+558T+p3T2 1 + 558 T + p^{3} T^{2}
59 1−540T+p3T2 1 - 540 T + p^{3} T^{2}
61 1+110T+p3T2 1 + 110 T + p^{3} T^{2}
67 1−140T+p3T2 1 - 140 T + p^{3} T^{2}
71 1−840T+p3T2 1 - 840 T + p^{3} T^{2}
73 1+550T+p3T2 1 + 550 T + p^{3} T^{2}
79 1−208T+p3T2 1 - 208 T + p^{3} T^{2}
83 1−516T+p3T2 1 - 516 T + p^{3} T^{2}
89 1+1398T+p3T2 1 + 1398 T + p^{3} T^{2}
97 1−1586T+p3T2 1 - 1586 T + p^{3} 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

−10.12528604205771189110666401735, −9.215823791984288944729899750293, −8.284646168257653410387205714789, −7.82442186692745137546595442365, −6.45479564422681832295287720107, −5.55314249092004707563323961682, −3.91205579060244468303258770839, −3.33920704810777841973751625046, −1.88179001471311342318053272741, 0, 1.88179001471311342318053272741, 3.33920704810777841973751625046, 3.91205579060244468303258770839, 5.55314249092004707563323961682, 6.45479564422681832295287720107, 7.82442186692745137546595442365, 8.284646168257653410387205714789, 9.215823791984288944729899750293, 10.12528604205771189110666401735

Graph of the ZZ-function along the critical line