Properties

Label 4-4707-1.1-c1e2-0-0
Degree $4$
Conductor $4707$
Sign $1$
Analytic cond. $0.300122$
Root an. cond. $0.740158$
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  − 2-s + 4-s + 2·5-s + 4·7-s − 3·8-s − 9-s − 2·10-s − 4·11-s − 13-s − 4·14-s + 16-s + 17-s + 18-s − 19-s + 2·20-s + 4·22-s − 23-s − 6·25-s + 26-s + 4·28-s + 3·29-s + 9·31-s + 32-s − 34-s + 8·35-s − 36-s − 2·37-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s + 1.51·7-s − 1.06·8-s − 1/3·9-s − 0.632·10-s − 1.20·11-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.229·19-s + 0.447·20-s + 0.852·22-s − 0.208·23-s − 6/5·25-s + 0.196·26-s + 0.755·28-s + 0.557·29-s + 1.61·31-s + 0.176·32-s − 0.171·34-s + 1.35·35-s − 1/6·36-s − 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4707 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4707 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4707\)    =    \(3^{2} \cdot 523\)
Sign: $1$
Analytic conductor: \(0.300122\)
Root analytic conductor: \(0.740158\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4707,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6921007709\)
\(L(\frac12)\) \(\approx\) \(0.6921007709\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
523$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
11$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T - 8 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T + 16 T^{2} - p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + T + 22 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 9 T + 46 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$D_{4}$ \( 1 - T + 72 T^{2} - p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 3 T + 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 3 T + 56 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 9 T + 158 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 7 T + 132 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T + 126 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−17.6446464738, −17.3943735887, −16.7948064293, −16.0263797797, −15.4939263621, −15.1822479007, −14.4723651596, −13.9799553447, −13.5653741706, −12.8743469278, −12.0276388975, −11.6883583204, −11.1621926186, −10.3566747483, −10.0572077670, −9.37557576083, −8.59565587401, −8.13171845431, −7.69024558721, −6.66541775562, −5.91565520192, −5.30877073338, −4.50542373379, −2.84785388566, −1.98472680966, 1.98472680966, 2.84785388566, 4.50542373379, 5.30877073338, 5.91565520192, 6.66541775562, 7.69024558721, 8.13171845431, 8.59565587401, 9.37557576083, 10.0572077670, 10.3566747483, 11.1621926186, 11.6883583204, 12.0276388975, 12.8743469278, 13.5653741706, 13.9799553447, 14.4723651596, 15.1822479007, 15.4939263621, 16.0263797797, 16.7948064293, 17.3943735887, 17.6446464738

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