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Properties

Label 2-447-447.446-c0-0-2
Degree $2$
Conductor $447$
Sign $1$
Analytic cond. $0.223082$
Root an. cond. $0.472315$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.80·2-s + 3-s + 2.24·4-s − 1.80·6-s − 0.445·7-s − 2.24·8-s + 9-s + 1.24·11-s + 2.24·12-s + 0.801·14-s + 1.80·16-s − 1.80·18-s − 1.80·19-s − 0.445·21-s − 2.24·22-s − 0.445·23-s − 2.24·24-s + 25-s + 27-s − 28-s + 1.24·31-s − 1.00·32-s + 1.24·33-s + 2.24·36-s − 1.80·37-s + 3.24·38-s + 1.24·41-s + ⋯
L(s)  = 1  − 1.80·2-s + 3-s + 2.24·4-s − 1.80·6-s − 0.445·7-s − 2.24·8-s + 9-s + 1.24·11-s + 2.24·12-s + 0.801·14-s + 1.80·16-s − 1.80·18-s − 1.80·19-s − 0.445·21-s − 2.24·22-s − 0.445·23-s − 2.24·24-s + 25-s + 27-s − 28-s + 1.24·31-s − 1.00·32-s + 1.24·33-s + 2.24·36-s − 1.80·37-s + 3.24·38-s + 1.24·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 447 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 447 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(447\)    =    \(3 \cdot 149\)
Sign: $1$
Analytic conductor: \(0.223082\)
Root analytic conductor: \(0.472315\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{447} (446, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 447,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5516413225\)
\(L(\frac12)\) \(\approx\) \(0.5516413225\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
149 \( 1 - T \)
good2 \( 1 + 1.80T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + 0.445T + T^{2} \)
11 \( 1 - 1.24T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.80T + T^{2} \)
23 \( 1 + 0.445T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.24T + T^{2} \)
37 \( 1 + 1.80T + T^{2} \)
41 \( 1 - 1.24T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.80T + T^{2} \)
61 \( 1 - 1.24T + T^{2} \)
67 \( 1 - 1.24T + T^{2} \)
71 \( 1 + 1.80T + T^{2} \)
73 \( 1 + 0.445T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 0.445T + T^{2} \)
89 \( 1 + 0.445T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.83825980316675005604832603592, −10.14866620493946758315578061031, −9.307110975782005013404183436806, −8.724622655345588025712028427955, −8.077015532215392242406297269297, −6.89583229976468280121918581623, −6.43635493766420963173751838123, −4.16629376621955412865319929053, −2.78319217042432333593650012333, −1.56232957238509874094185475666, 1.56232957238509874094185475666, 2.78319217042432333593650012333, 4.16629376621955412865319929053, 6.43635493766420963173751838123, 6.89583229976468280121918581623, 8.077015532215392242406297269297, 8.724622655345588025712028427955, 9.307110975782005013404183436806, 10.14866620493946758315578061031, 10.83825980316675005604832603592

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