Properties

Label 4-504-1.1-c1e2-0-0
Degree $4$
Conductor $504$
Sign $1$
Analytic cond. $0.0321354$
Root an. cond. $0.423395$
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-s − 4-s − 4·5-s − 7-s + 3·8-s + 9-s + 4·10-s + 8·11-s − 4·13-s + 14-s − 16-s − 4·17-s − 18-s + 4·20-s − 8·22-s − 8·23-s + 2·25-s + 4·26-s + 28-s + 4·29-s + 8·31-s − 5·32-s + 4·34-s + 4·35-s − 36-s + 12·37-s − 12·40-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 1.78·5-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 1.26·10-s + 2.41·11-s − 1.10·13-s + 0.267·14-s − 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.894·20-s − 1.70·22-s − 1.66·23-s + 2/5·25-s + 0.784·26-s + 0.188·28-s + 0.742·29-s + 1.43·31-s − 0.883·32-s + 0.685·34-s + 0.676·35-s − 1/6·36-s + 1.97·37-s − 1.89·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.0321354\)
Root analytic conductor: \(0.423395\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{504} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 504,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2432093576\)
\(L(\frac12)\) \(\approx\) \(0.2432093576\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−19.8970146357, −19.8934316645, −19.6678488176, −19.1551537949, −18.4192282680, −17.6707519500, −17.1598241820, −16.6272610009, −15.8734598001, −15.4732516475, −14.5135323404, −14.1819882616, −13.1937925053, −12.3172568607, −11.5820474614, −11.5559508175, −9.96467191573, −9.72466121073, −8.67236519668, −8.09899069409, −7.25047783803, −6.42897107073, −4.25303028693, −4.13559084051, 4.13559084051, 4.25303028693, 6.42897107073, 7.25047783803, 8.09899069409, 8.67236519668, 9.72466121073, 9.96467191573, 11.5559508175, 11.5820474614, 12.3172568607, 13.1937925053, 14.1819882616, 14.5135323404, 15.4732516475, 15.8734598001, 16.6272610009, 17.1598241820, 17.6707519500, 18.4192282680, 19.1551537949, 19.6678488176, 19.8934316645, 19.8970146357

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