Properties

Label 4-2009e2-1.1-c1e2-0-1
Degree $4$
Conductor $4036081$
Sign $1$
Analytic cond. $257.344$
Root an. cond. $4.00523$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·4-s + 5-s − 2·9-s − 5·11-s + 4·12-s + 13-s − 15-s + 12·16-s − 3·17-s − 4·20-s − 4·23-s − 6·25-s + 2·27-s + 4·29-s + 31-s + 5·33-s + 8·36-s − 39-s + 2·41-s − 12·43-s + 20·44-s − 2·45-s + 18·47-s − 12·48-s + 3·51-s − 4·52-s + ⋯
L(s)  = 1  − 0.577·3-s − 2·4-s + 0.447·5-s − 2/3·9-s − 1.50·11-s + 1.15·12-s + 0.277·13-s − 0.258·15-s + 3·16-s − 0.727·17-s − 0.894·20-s − 0.834·23-s − 6/5·25-s + 0.384·27-s + 0.742·29-s + 0.179·31-s + 0.870·33-s + 4/3·36-s − 0.160·39-s + 0.312·41-s − 1.82·43-s + 3.01·44-s − 0.298·45-s + 2.62·47-s − 1.73·48-s + 0.420·51-s − 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4036081 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4036081 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4036081\)    =    \(7^{4} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(257.344\)
Root analytic conductor: \(4.00523\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4036081,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad7 \( 1 \)
41$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
3$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 4 T + 49 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - T + 59 T^{2} - p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
53$C_4$ \( 1 - 9 T + 97 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 121 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 6 T + 130 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 15 T + 199 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 6 T - 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 16 T + 217 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 8 T + 181 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 15 T + 247 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
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

−8.964553996192192304835834185381, −8.584337313748584171008274975841, −8.292461396959523451698842973673, −8.011964882236786287447004917189, −7.30765305099540558791090349400, −7.30630530579428335913130207460, −6.27324694550977427497958356907, −5.96693530228955912877585919749, −5.64135080796154896978556221998, −5.51138324868263757577110361822, −4.79040531856481382664727393853, −4.67960341094204373319430849009, −4.04935865377997381264276759061, −3.78605279638168857222091759104, −2.97327408335083045607922851207, −2.66115787201260131618437153258, −1.86748536113438001741414829712, −1.04438508788416614721215932322, 0, 0, 1.04438508788416614721215932322, 1.86748536113438001741414829712, 2.66115787201260131618437153258, 2.97327408335083045607922851207, 3.78605279638168857222091759104, 4.04935865377997381264276759061, 4.67960341094204373319430849009, 4.79040531856481382664727393853, 5.51138324868263757577110361822, 5.64135080796154896978556221998, 5.96693530228955912877585919749, 6.27324694550977427497958356907, 7.30630530579428335913130207460, 7.30765305099540558791090349400, 8.011964882236786287447004917189, 8.292461396959523451698842973673, 8.584337313748584171008274975841, 8.964553996192192304835834185381

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