Properties

Label 4-10005-1.1-c1e2-0-0
Degree $4$
Conductor $10005$
Sign $-1$
Analytic cond. $0.637927$
Root an. cond. $0.893702$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·4-s − 5-s + 7-s + 3·8-s − 2·9-s + 10-s − 3·11-s − 14-s + 16-s − 3·17-s + 2·18-s − 6·19-s + 2·20-s + 3·22-s + 3·23-s − 2·25-s + 3·27-s − 2·28-s − 17·31-s − 2·32-s + 3·34-s − 35-s + 4·36-s + 6·38-s − 3·40-s − 7·41-s + ⋯
L(s)  = 1  − 0.707·2-s − 4-s − 0.447·5-s + 0.377·7-s + 1.06·8-s − 2/3·9-s + 0.316·10-s − 0.904·11-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 1.37·19-s + 0.447·20-s + 0.639·22-s + 0.625·23-s − 2/5·25-s + 0.577·27-s − 0.377·28-s − 3.05·31-s − 0.353·32-s + 0.514·34-s − 0.169·35-s + 2/3·36-s + 0.973·38-s − 0.474·40-s − 1.09·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10005\)    =    \(3 \cdot 5 \cdot 23 \cdot 29\)
Sign: $-1$
Analytic conductor: \(0.637927\)
Root analytic conductor: \(0.893702\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10005} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 10005,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
29$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 12 T + 101 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$D_{4}$ \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$C_4$ \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 3 T + 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$D_{4}$ \( 1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 21 T + 280 T^{2} - 21 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 10 T + 86 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.9022520408, −16.6033486419, −15.8750911565, −15.3589801706, −14.7915586251, −14.5299799519, −13.7423862609, −13.4003262242, −12.8760249759, −12.4180026791, −11.7043487417, −10.9490184874, −10.7186060238, −10.2238135607, −9.18240340140, −8.98322378679, −8.60430763895, −7.85344423968, −7.48661667188, −6.57461008972, −5.63570310853, −5.01331521552, −4.34894466000, −3.52167359876, −2.19061595016, 0, 2.19061595016, 3.52167359876, 4.34894466000, 5.01331521552, 5.63570310853, 6.57461008972, 7.48661667188, 7.85344423968, 8.60430763895, 8.98322378679, 9.18240340140, 10.2238135607, 10.7186060238, 10.9490184874, 11.7043487417, 12.4180026791, 12.8760249759, 13.4003262242, 13.7423862609, 14.5299799519, 14.7915586251, 15.3589801706, 15.8750911565, 16.6033486419, 16.9022520408

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