Properties

Label 4-1660608-1.1-c1e2-0-0
Degree $4$
Conductor $1660608$
Sign $1$
Analytic cond. $105.881$
Root an. cond. $3.20778$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 8·11-s − 4·17-s − 8·19-s + 16·23-s − 6·25-s + 27-s − 12·29-s + 8·31-s − 8·33-s − 14·49-s − 4·51-s + 4·53-s − 8·57-s − 8·67-s + 16·69-s − 6·75-s + 81-s + 8·83-s − 12·87-s + 12·89-s + 8·93-s + 4·97-s − 8·99-s + 32·103-s − 4·109-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 2.41·11-s − 0.970·17-s − 1.83·19-s + 3.33·23-s − 6/5·25-s + 0.192·27-s − 2.22·29-s + 1.43·31-s − 1.39·33-s − 2·49-s − 0.560·51-s + 0.549·53-s − 1.05·57-s − 0.977·67-s + 1.92·69-s − 0.692·75-s + 1/9·81-s + 0.878·83-s − 1.28·87-s + 1.27·89-s + 0.829·93-s + 0.406·97-s − 0.804·99-s + 3.15·103-s − 0.383·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1660608 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1660608 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1660608\)    =    \(2^{6} \cdot 3^{3} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(105.881\)
Root analytic conductor: \(3.20778\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1660608} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1660608,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.178704757\)
\(L(\frac12)\) \(\approx\) \(1.178704757\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3$C_1$ \( 1 - T \)
31$C_2$ \( 1 - 8 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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

−7.70160423858248593329309475413, −7.62403207119791885504595168937, −7.10071299276536368111515418477, −6.62856063110233567525397669622, −6.14969218484336320739052245813, −5.69170053737875154299643805931, −5.00834690207985129093078924318, −4.83326855247570334999188527992, −4.46842829081039592357457394445, −3.60460660885553352957233947063, −3.26022314787984145251855830027, −2.51659494204614894722398853037, −2.36656187810192615068409367919, −1.65372841158199596174232192019, −0.41421208904226405922670150583, 0.41421208904226405922670150583, 1.65372841158199596174232192019, 2.36656187810192615068409367919, 2.51659494204614894722398853037, 3.26022314787984145251855830027, 3.60460660885553352957233947063, 4.46842829081039592357457394445, 4.83326855247570334999188527992, 5.00834690207985129093078924318, 5.69170053737875154299643805931, 6.14969218484336320739052245813, 6.62856063110233567525397669622, 7.10071299276536368111515418477, 7.62403207119791885504595168937, 7.70160423858248593329309475413

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