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Properties

Label 2-448-1.1-c5-0-31
Degree 22
Conductor 448448
Sign −1-1
Analytic cond. 71.851971.8519
Root an. cond. 8.476558.47655
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.52·3-s − 72.6·5-s − 49·7-s − 186.·9-s − 30.4·11-s + 1.14e3·13-s + 546.·15-s − 514.·17-s + 2.31e3·19-s + 368.·21-s − 409.·23-s + 2.15e3·25-s + 3.23e3·27-s + 1.69e3·29-s + 7.03e3·31-s + 229.·33-s + 3.55e3·35-s − 3.93e3·37-s − 8.61e3·39-s − 9.23e3·41-s − 2.13e4·43-s + 1.35e4·45-s − 9.94e3·47-s + 2.40e3·49-s + 3.86e3·51-s + 3.45e4·53-s + 2.21e3·55-s + ⋯
L(s)  = 1  − 0.482·3-s − 1.29·5-s − 0.377·7-s − 0.767·9-s − 0.0758·11-s + 1.88·13-s + 0.626·15-s − 0.431·17-s + 1.47·19-s + 0.182·21-s − 0.161·23-s + 0.688·25-s + 0.852·27-s + 0.373·29-s + 1.31·31-s + 0.0366·33-s + 0.491·35-s − 0.472·37-s − 0.907·39-s − 0.858·41-s − 1.76·43-s + 0.996·45-s − 0.656·47-s + 0.142·49-s + 0.208·51-s + 1.69·53-s + 0.0985·55-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: −1-1
Analytic conductor: 71.851971.8519
Root analytic conductor: 8.476558.47655
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 448, ( :5/2), −1)(2,\ 448,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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+49T 1 + 49T
good3 1+7.52T+243T2 1 + 7.52T + 243T^{2}
5 1+72.6T+3.12e3T2 1 + 72.6T + 3.12e3T^{2}
11 1+30.4T+1.61e5T2 1 + 30.4T + 1.61e5T^{2}
13 1−1.14e3T+3.71e5T2 1 - 1.14e3T + 3.71e5T^{2}
17 1+514.T+1.41e6T2 1 + 514.T + 1.41e6T^{2}
19 1−2.31e3T+2.47e6T2 1 - 2.31e3T + 2.47e6T^{2}
23 1+409.T+6.43e6T2 1 + 409.T + 6.43e6T^{2}
29 1−1.69e3T+2.05e7T2 1 - 1.69e3T + 2.05e7T^{2}
31 1−7.03e3T+2.86e7T2 1 - 7.03e3T + 2.86e7T^{2}
37 1+3.93e3T+6.93e7T2 1 + 3.93e3T + 6.93e7T^{2}
41 1+9.23e3T+1.15e8T2 1 + 9.23e3T + 1.15e8T^{2}
43 1+2.13e4T+1.47e8T2 1 + 2.13e4T + 1.47e8T^{2}
47 1+9.94e3T+2.29e8T2 1 + 9.94e3T + 2.29e8T^{2}
53 1−3.45e4T+4.18e8T2 1 - 3.45e4T + 4.18e8T^{2}
59 1+4.63e4T+7.14e8T2 1 + 4.63e4T + 7.14e8T^{2}
61 1−3.38e4T+8.44e8T2 1 - 3.38e4T + 8.44e8T^{2}
67 1−4.92e4T+1.35e9T2 1 - 4.92e4T + 1.35e9T^{2}
71 1+4.95e4T+1.80e9T2 1 + 4.95e4T + 1.80e9T^{2}
73 1−3.38e4T+2.07e9T2 1 - 3.38e4T + 2.07e9T^{2}
79 1−622.T+3.07e9T2 1 - 622.T + 3.07e9T^{2}
83 1+9.72e4T+3.93e9T2 1 + 9.72e4T + 3.93e9T^{2}
89 1+1.69e4T+5.58e9T2 1 + 1.69e4T + 5.58e9T^{2}
97 1+6.54e4T+8.58e9T2 1 + 6.54e4T + 8.58e9T^{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

−9.978230480836902827785022286177, −8.608526818452706838020256589848, −8.233031539052172044838508921563, −6.97350018624386768440627804046, −6.12176068106607292035029742376, −5.05333174841542167849968857730, −3.80155628252662423528086357564, −3.07930539757161754397212343379, −1.08409089898978280261779642416, 0, 1.08409089898978280261779642416, 3.07930539757161754397212343379, 3.80155628252662423528086357564, 5.05333174841542167849968857730, 6.12176068106607292035029742376, 6.97350018624386768440627804046, 8.233031539052172044838508921563, 8.608526818452706838020256589848, 9.978230480836902827785022286177

Graph of the ZZ-function along the critical line