Properties

Label 4-22e2-1.1-c1e2-0-0
Degree $4$
Conductor $484$
Sign $1$
Analytic cond. $0.0308602$
Root an. cond. $0.419131$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 2·5-s − 5·9-s + 4·10-s − 4·16-s + 4·17-s + 10·18-s + 8·19-s − 4·20-s − 4·23-s − 3·25-s + 12·31-s + 8·32-s − 8·34-s − 10·36-s + 2·37-s − 16·38-s − 8·41-s − 16·43-s + 10·45-s + 8·46-s + 8·47-s − 10·49-s + 6·50-s − 12·53-s + 8·59-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.894·5-s − 5/3·9-s + 1.26·10-s − 16-s + 0.970·17-s + 2.35·18-s + 1.83·19-s − 0.894·20-s − 0.834·23-s − 3/5·25-s + 2.15·31-s + 1.41·32-s − 1.37·34-s − 5/3·36-s + 0.328·37-s − 2.59·38-s − 1.24·41-s − 2.43·43-s + 1.49·45-s + 1.17·46-s + 1.16·47-s − 1.42·49-s + 0.848·50-s − 1.64·53-s + 1.04·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.0308602\)
Root analytic conductor: \(0.419131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{484} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 484,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2042529109\)
\(L(\frac12)\) \(\approx\) \(0.2042529109\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + p T + p T^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.3922718580, −19.1857249719, −18.2764512729, −17.9414335735, −17.0336103204, −16.9483853593, −15.9140726033, −15.7772041936, −14.7397148174, −13.9818243124, −13.5686390571, −12.1137747715, −11.5293266281, −11.4512586103, −10.0355090972, −9.81375356518, −8.60353961929, −8.13266580748, −7.60852684016, −6.36261389471, −5.11227132196, −3.27381985667, 3.27381985667, 5.11227132196, 6.36261389471, 7.60852684016, 8.13266580748, 8.60353961929, 9.81375356518, 10.0355090972, 11.4512586103, 11.5293266281, 12.1137747715, 13.5686390571, 13.9818243124, 14.7397148174, 15.7772041936, 15.9140726033, 16.9483853593, 17.0336103204, 17.9414335735, 18.2764512729, 19.1857249719, 19.3922718580

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