Properties

Label 4-209e2-1.1-c1e2-0-2
Degree $4$
Conductor $43681$
Sign $1$
Analytic cond. $2.78513$
Root an. cond. $1.29184$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·4-s − 2·5-s − 4·7-s − 9-s − 2·11-s + 4·12-s − 4·13-s + 4·15-s + 4·17-s − 2·19-s + 4·20-s + 8·21-s − 6·23-s − 7·25-s + 6·27-s + 8·28-s − 4·29-s − 10·31-s + 4·33-s + 8·35-s + 2·36-s + 6·37-s + 8·39-s + 8·41-s + 12·43-s + 4·44-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 0.894·5-s − 1.51·7-s − 1/3·9-s − 0.603·11-s + 1.15·12-s − 1.10·13-s + 1.03·15-s + 0.970·17-s − 0.458·19-s + 0.894·20-s + 1.74·21-s − 1.25·23-s − 7/5·25-s + 1.15·27-s + 1.51·28-s − 0.742·29-s − 1.79·31-s + 0.696·33-s + 1.35·35-s + 1/3·36-s + 0.986·37-s + 1.28·39-s + 1.24·41-s + 1.82·43-s + 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(43681\)    =    \(11^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(2.78513\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 43681,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad11$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 4 T + 44 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 10 T + 85 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 33 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 125 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 88 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 18 T + 213 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 22 T + 261 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 32 T + 412 T^{2} + 32 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 168 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 10 T + 105 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 2 T + 193 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−12.07190025498865328151400289003, −11.78528494648351285918908043889, −11.27778035721523365343604018422, −10.67603568548032238096009848013, −10.17657948352155952925893447062, −9.760607320506683258382618585141, −9.121411903582605984216326523335, −8.961927499800109227389829265358, −7.942479983460655289164991279617, −7.46822210305757932839889624254, −7.29991824030977858611988412521, −5.99919994480041418957486446423, −5.85127016788906070010672256042, −5.58376600789641197137620065565, −4.37512198625136697918539487478, −4.22551461551423115358213443029, −3.33673779054846715421052009264, −2.48727755494252447957045829685, 0, 0, 2.48727755494252447957045829685, 3.33673779054846715421052009264, 4.22551461551423115358213443029, 4.37512198625136697918539487478, 5.58376600789641197137620065565, 5.85127016788906070010672256042, 5.99919994480041418957486446423, 7.29991824030977858611988412521, 7.46822210305757932839889624254, 7.942479983460655289164991279617, 8.961927499800109227389829265358, 9.121411903582605984216326523335, 9.760607320506683258382618585141, 10.17657948352155952925893447062, 10.67603568548032238096009848013, 11.27778035721523365343604018422, 11.78528494648351285918908043889, 12.07190025498865328151400289003

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