Properties

Label 4-1008e2-1.1-c1e2-0-139
Degree $4$
Conductor $1016064$
Sign $1$
Analytic cond. $64.7851$
Root an. cond. $2.83706$
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  − 3·3-s − 7-s + 6·9-s − 6·11-s − 3·13-s + 3·17-s − 5·19-s + 3·21-s + 6·23-s − 5·25-s − 9·27-s − 3·29-s + 2·31-s + 18·33-s − 7·37-s + 9·39-s − 3·41-s + 3·43-s − 18·47-s − 6·49-s − 9·51-s + 9·53-s + 15·57-s − 30·59-s − 6·63-s − 18·69-s + 3·73-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.377·7-s + 2·9-s − 1.80·11-s − 0.832·13-s + 0.727·17-s − 1.14·19-s + 0.654·21-s + 1.25·23-s − 25-s − 1.73·27-s − 0.557·29-s + 0.359·31-s + 3.13·33-s − 1.15·37-s + 1.44·39-s − 0.468·41-s + 0.457·43-s − 2.62·47-s − 6/7·49-s − 1.26·51-s + 1.23·53-s + 1.98·57-s − 3.90·59-s − 0.755·63-s − 2.16·69-s + 0.351·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(64.7851\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1008} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1016064,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2 \( 1 \)
3$C_2$ \( 1 + p T + p T^{2} \)
7$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 3 T + 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 119 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 155 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 3 T + 92 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 3 T + 100 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
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

−9.771723215978993452367705885833, −9.656113499038128208853579956664, −9.088472285284999150630395869925, −8.334228411483683157152767648286, −8.001353158581588686890231171607, −7.64485300205579036115538510653, −6.95427890014940484405631502180, −6.94572249183600658345164168460, −6.14341443072050005131946147359, −5.98103318494248854148557399498, −5.22125234839579588255596246994, −5.19968429157954576102283825472, −4.67458007972770686813472574152, −4.21539192318077838226345494506, −3.30297762789591676320546996660, −2.94806202445393416291259000576, −2.07987778364118918133099247406, −1.40780492243388931553261632538, 0, 0, 1.40780492243388931553261632538, 2.07987778364118918133099247406, 2.94806202445393416291259000576, 3.30297762789591676320546996660, 4.21539192318077838226345494506, 4.67458007972770686813472574152, 5.19968429157954576102283825472, 5.22125234839579588255596246994, 5.98103318494248854148557399498, 6.14341443072050005131946147359, 6.94572249183600658345164168460, 6.95427890014940484405631502180, 7.64485300205579036115538510653, 8.001353158581588686890231171607, 8.334228411483683157152767648286, 9.088472285284999150630395869925, 9.656113499038128208853579956664, 9.771723215978993452367705885833

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