Properties

Label 2-448-1.1-c5-0-0
Degree $2$
Conductor $448$
Sign $1$
Analytic cond. $71.8519$
Root an. cond. $8.47655$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 25.6·3-s − 28.7·5-s − 49·7-s + 414.·9-s − 270.·11-s − 300.·13-s + 737.·15-s + 613.·17-s − 1.70e3·19-s + 1.25e3·21-s − 3.18e3·23-s − 2.29e3·25-s − 4.40e3·27-s − 4.29e3·29-s − 2.02e3·31-s + 6.92e3·33-s + 1.40e3·35-s − 5.15e3·37-s + 7.71e3·39-s − 7.14e3·41-s − 1.95e4·43-s − 1.19e4·45-s − 1.99e4·47-s + 2.40e3·49-s − 1.57e4·51-s − 3.94e3·53-s + 7.76e3·55-s + ⋯
L(s)  = 1  − 1.64·3-s − 0.514·5-s − 0.377·7-s + 1.70·9-s − 0.673·11-s − 0.493·13-s + 0.846·15-s + 0.514·17-s − 1.08·19-s + 0.621·21-s − 1.25·23-s − 0.735·25-s − 1.16·27-s − 0.949·29-s − 0.379·31-s + 1.10·33-s + 0.194·35-s − 0.618·37-s + 0.811·39-s − 0.663·41-s − 1.61·43-s − 0.878·45-s − 1.32·47-s + 0.142·49-s − 0.846·51-s − 0.193·53-s + 0.346·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $1$
Analytic conductor: \(71.8519\)
Root analytic conductor: \(8.47655\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.04845593002\)
\(L(\frac12)\) \(\approx\) \(0.04845593002\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + 49T \)
good3 \( 1 + 25.6T + 243T^{2} \)
5 \( 1 + 28.7T + 3.12e3T^{2} \)
11 \( 1 + 270.T + 1.61e5T^{2} \)
13 \( 1 + 300.T + 3.71e5T^{2} \)
17 \( 1 - 613.T + 1.41e6T^{2} \)
19 \( 1 + 1.70e3T + 2.47e6T^{2} \)
23 \( 1 + 3.18e3T + 6.43e6T^{2} \)
29 \( 1 + 4.29e3T + 2.05e7T^{2} \)
31 \( 1 + 2.02e3T + 2.86e7T^{2} \)
37 \( 1 + 5.15e3T + 6.93e7T^{2} \)
41 \( 1 + 7.14e3T + 1.15e8T^{2} \)
43 \( 1 + 1.95e4T + 1.47e8T^{2} \)
47 \( 1 + 1.99e4T + 2.29e8T^{2} \)
53 \( 1 + 3.94e3T + 4.18e8T^{2} \)
59 \( 1 + 2.97e4T + 7.14e8T^{2} \)
61 \( 1 - 5.05e4T + 8.44e8T^{2} \)
67 \( 1 - 5.05e3T + 1.35e9T^{2} \)
71 \( 1 + 3.28e4T + 1.80e9T^{2} \)
73 \( 1 + 1.11e4T + 2.07e9T^{2} \)
79 \( 1 + 8.18e4T + 3.07e9T^{2} \)
83 \( 1 - 1.18e5T + 3.93e9T^{2} \)
89 \( 1 + 4.16e4T + 5.58e9T^{2} \)
97 \( 1 - 4.36e4T + 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.34764045866613352589988820276, −9.837500151823912601639328628504, −8.305204492276353892197827127686, −7.35243339953161013876354676592, −6.41211315047822079728635977487, −5.59385894006586569909427544954, −4.72219214272524065445870736166, −3.61333479769640992199747184961, −1.84414697699420323307111578254, −0.11883944131619943546311095232, 0.11883944131619943546311095232, 1.84414697699420323307111578254, 3.61333479769640992199747184961, 4.72219214272524065445870736166, 5.59385894006586569909427544954, 6.41211315047822079728635977487, 7.35243339953161013876354676592, 8.305204492276353892197827127686, 9.837500151823912601639328628504, 10.34764045866613352589988820276

Graph of the $Z$-function along the critical line