Properties

Label 4-4617-1.1-c1e2-0-0
Degree $4$
Conductor $4617$
Sign $1$
Analytic cond. $0.294383$
Root an. cond. $0.736594$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4-s + 7-s + 9-s − 3·11-s − 12-s + 4·13-s − 3·16-s − 3·17-s − 3·19-s + 21-s + 3·23-s + 2·25-s + 27-s − 28-s − 6·29-s − 2·31-s − 3·33-s − 36-s − 2·37-s + 4·39-s + 6·41-s + 7·43-s + 3·44-s − 15·47-s − 3·48-s − 5·49-s + ⋯
L(s)  = 1  + 0.577·3-s − 1/2·4-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 1.10·13-s − 3/4·16-s − 0.727·17-s − 0.688·19-s + 0.218·21-s + 0.625·23-s + 2/5·25-s + 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.359·31-s − 0.522·33-s − 1/6·36-s − 0.328·37-s + 0.640·39-s + 0.937·41-s + 1.06·43-s + 0.452·44-s − 2.18·47-s − 0.433·48-s − 5/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4617 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4617 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4617\)    =    \(3^{5} \cdot 19\)
Sign: $1$
Analytic conductor: \(0.294383\)
Root analytic conductor: \(0.736594\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4617,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8777387316\)
\(L(\frac12)\) \(\approx\) \(0.8777387316\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( 1 - T \)
19$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$D_{4}$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$D_{4}$ \( 1 - 7 T + 66 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 15 T + 118 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 142 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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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

−17.7385697581, −17.2826677364, −16.4632914025, −15.9429838371, −15.6693246722, −14.9542396189, −14.4767732915, −14.0445824845, −13.2181860468, −13.0372798974, −12.7565450689, −11.5176261395, −11.1187636010, −10.7353858957, −9.84969756358, −9.25957793425, −8.61904573850, −8.32618530262, −7.48177885459, −6.78099269333, −5.93419446583, −4.98268675734, −4.33592291464, −3.36319567894, −2.11290454270, 2.11290454270, 3.36319567894, 4.33592291464, 4.98268675734, 5.93419446583, 6.78099269333, 7.48177885459, 8.32618530262, 8.61904573850, 9.25957793425, 9.84969756358, 10.7353858957, 11.1187636010, 11.5176261395, 12.7565450689, 13.0372798974, 13.2181860468, 14.0445824845, 14.4767732915, 14.9542396189, 15.6693246722, 15.9429838371, 16.4632914025, 17.2826677364, 17.7385697581

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