Properties

Label 4-1280e2-1.1-c1e2-0-38
Degree $4$
Conductor $1638400$
Sign $1$
Analytic cond. $104.465$
Root an. cond. $3.19700$
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  − 4·3-s + 4·5-s + 6·9-s + 8·13-s − 16·15-s + 11·25-s + 4·27-s + 8·37-s − 32·39-s − 4·41-s − 12·43-s + 24·45-s + 10·49-s + 8·53-s + 32·65-s + 28·67-s + 16·71-s − 44·75-s + 32·79-s − 37·81-s − 4·83-s + 12·89-s + 20·107-s − 32·111-s + 48·117-s + 6·121-s + 16·123-s + ⋯
L(s)  = 1  − 2.30·3-s + 1.78·5-s + 2·9-s + 2.21·13-s − 4.13·15-s + 11/5·25-s + 0.769·27-s + 1.31·37-s − 5.12·39-s − 0.624·41-s − 1.82·43-s + 3.57·45-s + 10/7·49-s + 1.09·53-s + 3.96·65-s + 3.42·67-s + 1.89·71-s − 5.08·75-s + 3.60·79-s − 4.11·81-s − 0.439·83-s + 1.27·89-s + 1.93·107-s − 3.03·111-s + 4.43·117-s + 6/11·121-s + 1.44·123-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1638400\)    =    \(2^{16} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(104.465\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1638400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.722474975\)
\(L(\frac12)\) \(\approx\) \(1.722474975\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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

−9.928971189577272098083152588991, −9.766372745339745560726844676196, −8.945625355400746826861190941885, −8.899684413635341815554324519298, −8.254349233110019116551708517282, −7.995872186926222820887291318071, −6.85510575013867939207248427383, −6.83149920291788978955989827861, −6.31347141187864079503047202649, −6.15266795230675776870945953065, −5.73312540063411365218110417729, −5.42189150993244898649232353874, −4.98886712055636563781281136304, −4.73806583088131163762078593581, −3.72014476271699181405675294422, −3.49200426452174103742797152596, −2.45651484586877876839425457570, −2.01036711970924889086730892421, −0.949837446179303178736793803052, −0.903086652817957475459194413825, 0.903086652817957475459194413825, 0.949837446179303178736793803052, 2.01036711970924889086730892421, 2.45651484586877876839425457570, 3.49200426452174103742797152596, 3.72014476271699181405675294422, 4.73806583088131163762078593581, 4.98886712055636563781281136304, 5.42189150993244898649232353874, 5.73312540063411365218110417729, 6.15266795230675776870945953065, 6.31347141187864079503047202649, 6.83149920291788978955989827861, 6.85510575013867939207248427383, 7.995872186926222820887291318071, 8.254349233110019116551708517282, 8.899684413635341815554324519298, 8.945625355400746826861190941885, 9.766372745339745560726844676196, 9.928971189577272098083152588991

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