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Properties

Label 2-448-1.1-c5-0-9
Degree 22
Conductor 448448
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.304·3-s − 35.5·5-s − 49·7-s − 242.·9-s + 565.·11-s − 983.·13-s − 10.8·15-s + 200.·17-s + 828.·19-s − 14.9·21-s − 4.43e3·23-s − 1.86e3·25-s − 147.·27-s + 3.71e3·29-s − 992.·31-s + 172.·33-s + 1.74e3·35-s + 8.35e3·37-s − 299.·39-s − 1.34e4·41-s + 298.·43-s + 8.62e3·45-s + 1.87e4·47-s + 2.40e3·49-s + 60.8·51-s − 1.60e4·53-s − 2.00e4·55-s + ⋯
L(s)  = 1  + 0.0195·3-s − 0.635·5-s − 0.377·7-s − 0.999·9-s + 1.40·11-s − 1.61·13-s − 0.0123·15-s + 0.167·17-s + 0.526·19-s − 0.00737·21-s − 1.74·23-s − 0.596·25-s − 0.0390·27-s + 0.820·29-s − 0.185·31-s + 0.0275·33-s + 0.240·35-s + 1.00·37-s − 0.0314·39-s − 1.25·41-s + 0.0246·43-s + 0.635·45-s + 1.23·47-s + 0.142·49-s + 0.00327·51-s − 0.784·53-s − 0.895·55-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: 11
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: 00
Selberg data: (2, 448, ( :5/2), 1)(2,\ 448,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 1.1111935571.111193557
L(12)L(\frac12) ≈\approx 1.1111935571.111193557
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 1−0.304T+243T2 1 - 0.304T + 243T^{2}
5 1+35.5T+3.12e3T2 1 + 35.5T + 3.12e3T^{2}
11 1−565.T+1.61e5T2 1 - 565.T + 1.61e5T^{2}
13 1+983.T+3.71e5T2 1 + 983.T + 3.71e5T^{2}
17 1−200.T+1.41e6T2 1 - 200.T + 1.41e6T^{2}
19 1−828.T+2.47e6T2 1 - 828.T + 2.47e6T^{2}
23 1+4.43e3T+6.43e6T2 1 + 4.43e3T + 6.43e6T^{2}
29 1−3.71e3T+2.05e7T2 1 - 3.71e3T + 2.05e7T^{2}
31 1+992.T+2.86e7T2 1 + 992.T + 2.86e7T^{2}
37 1−8.35e3T+6.93e7T2 1 - 8.35e3T + 6.93e7T^{2}
41 1+1.34e4T+1.15e8T2 1 + 1.34e4T + 1.15e8T^{2}
43 1−298.T+1.47e8T2 1 - 298.T + 1.47e8T^{2}
47 1−1.87e4T+2.29e8T2 1 - 1.87e4T + 2.29e8T^{2}
53 1+1.60e4T+4.18e8T2 1 + 1.60e4T + 4.18e8T^{2}
59 1−1.27e4T+7.14e8T2 1 - 1.27e4T + 7.14e8T^{2}
61 1−3.49e4T+8.44e8T2 1 - 3.49e4T + 8.44e8T^{2}
67 1−1.19e4T+1.35e9T2 1 - 1.19e4T + 1.35e9T^{2}
71 1−1.29e4T+1.80e9T2 1 - 1.29e4T + 1.80e9T^{2}
73 1−8.11e4T+2.07e9T2 1 - 8.11e4T + 2.07e9T^{2}
79 1+4.69e4T+3.07e9T2 1 + 4.69e4T + 3.07e9T^{2}
83 1+1.11e5T+3.93e9T2 1 + 1.11e5T + 3.93e9T^{2}
89 1+3.47e4T+5.58e9T2 1 + 3.47e4T + 5.58e9T^{2}
97 1−9.26e4T+8.58e9T2 1 - 9.26e4T + 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

−10.09722694272163780500435925270, −9.484110496366488042071638998902, −8.434913414151653896298519983112, −7.58675218674520602490354732949, −6.59478390328425456077222121689, −5.61560310304633749081479944148, −4.36394503355686493807207108902, −3.41404759561890926887120510069, −2.20470176531966949214438510023, −0.52178033605158052394811031589, 0.52178033605158052394811031589, 2.20470176531966949214438510023, 3.41404759561890926887120510069, 4.36394503355686493807207108902, 5.61560310304633749081479944148, 6.59478390328425456077222121689, 7.58675218674520602490354732949, 8.434913414151653896298519983112, 9.484110496366488042071638998902, 10.09722694272163780500435925270

Graph of the ZZ-function along the critical line