Properties

Label 2-448-1.1-c5-0-41
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  + 30.3·3-s + 100.·5-s − 49·7-s + 676.·9-s + 140.·11-s − 725.·13-s + 3.04e3·15-s − 365.·17-s + 202.·19-s − 1.48e3·21-s + 2.73e3·23-s + 6.95e3·25-s + 1.31e4·27-s − 2.17e3·29-s − 8.95e3·31-s + 4.25e3·33-s − 4.91e3·35-s + 2.60e3·37-s − 2.19e4·39-s + 1.42e4·41-s + 1.12e4·43-s + 6.79e4·45-s + 1.77e4·47-s + 2.40e3·49-s − 1.10e4·51-s − 2.59e4·53-s + 1.40e4·55-s + ⋯
L(s)  = 1  + 1.94·3-s + 1.79·5-s − 0.377·7-s + 2.78·9-s + 0.349·11-s − 1.19·13-s + 3.49·15-s − 0.306·17-s + 0.128·19-s − 0.735·21-s + 1.07·23-s + 2.22·25-s + 3.47·27-s − 0.480·29-s − 1.67·31-s + 0.680·33-s − 0.678·35-s + 0.312·37-s − 2.31·39-s + 1.32·41-s + 0.926·43-s + 5.00·45-s + 1.17·47-s + 0.142·49-s − 0.597·51-s − 1.26·53-s + 0.628·55-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

\(L(3)\) \(\approx\) \(6.396469266\)
\(L(\frac12)\) \(\approx\) \(6.396469266\)
\(L(\frac{7}{2})\) 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 \( 1 - 30.3T + 243T^{2} \)
5 \( 1 - 100.T + 3.12e3T^{2} \)
11 \( 1 - 140.T + 1.61e5T^{2} \)
13 \( 1 + 725.T + 3.71e5T^{2} \)
17 \( 1 + 365.T + 1.41e6T^{2} \)
19 \( 1 - 202.T + 2.47e6T^{2} \)
23 \( 1 - 2.73e3T + 6.43e6T^{2} \)
29 \( 1 + 2.17e3T + 2.05e7T^{2} \)
31 \( 1 + 8.95e3T + 2.86e7T^{2} \)
37 \( 1 - 2.60e3T + 6.93e7T^{2} \)
41 \( 1 - 1.42e4T + 1.15e8T^{2} \)
43 \( 1 - 1.12e4T + 1.47e8T^{2} \)
47 \( 1 - 1.77e4T + 2.29e8T^{2} \)
53 \( 1 + 2.59e4T + 4.18e8T^{2} \)
59 \( 1 - 1.76e4T + 7.14e8T^{2} \)
61 \( 1 + 3.93e3T + 8.44e8T^{2} \)
67 \( 1 - 854.T + 1.35e9T^{2} \)
71 \( 1 + 1.91e3T + 1.80e9T^{2} \)
73 \( 1 + 4.88e4T + 2.07e9T^{2} \)
79 \( 1 + 5.55e4T + 3.07e9T^{2} \)
83 \( 1 + 9.76e4T + 3.93e9T^{2} \)
89 \( 1 + 3.14e4T + 5.58e9T^{2} \)
97 \( 1 + 2.96e4T + 8.58e9T^{2} \)
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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

−9.826625639943145665137765841128, −9.297195200760575119457915270731, −8.926463495479328924447188553538, −7.51286618139772598759450194646, −6.85930728161197779703455718506, −5.55329624771796622913512841619, −4.28260748774100532653656515142, −2.92836274895404229594944139727, −2.32071434533119790629198710139, −1.38107287485903568329374768848, 1.38107287485903568329374768848, 2.32071434533119790629198710139, 2.92836274895404229594944139727, 4.28260748774100532653656515142, 5.55329624771796622913512841619, 6.85930728161197779703455718506, 7.51286618139772598759450194646, 8.926463495479328924447188553538, 9.297195200760575119457915270731, 9.826625639943145665137765841128

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