Properties

Label 4-5376e2-1.1-c1e2-0-15
Degree $4$
Conductor $28901376$
Sign $1$
Analytic cond. $1842.77$
Root an. cond. $6.55191$
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  + 2·7-s − 9-s − 4·17-s + 8·23-s + 6·25-s + 16·31-s + 4·41-s + 3·49-s − 2·63-s − 24·71-s − 4·73-s + 81-s − 12·89-s + 4·97-s + 16·103-s − 28·113-s − 8·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 16·161-s + ⋯
L(s)  = 1  + 0.755·7-s − 1/3·9-s − 0.970·17-s + 1.66·23-s + 6/5·25-s + 2.87·31-s + 0.624·41-s + 3/7·49-s − 0.251·63-s − 2.84·71-s − 0.468·73-s + 1/9·81-s − 1.27·89-s + 0.406·97-s + 1.57·103-s − 2.63·113-s − 0.733·119-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.323·153-s + 0.0798·157-s + 1.26·161-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(28901376\)    =    \(2^{16} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1842.77\)
Root analytic conductor: \(6.55191\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5376} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 28901376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.043565475\)
\(L(\frac12)\) \(\approx\) \(3.043565475\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
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

−8.364146612788316031959783260767, −8.039410678429660950698247707711, −7.72408032474548797665328768976, −7.09105923397359768950600885078, −7.08490668393334585825389299390, −6.57003800194958448515494257296, −6.28592909289214005307878865614, −5.80683555839617801561362708889, −5.50892827791143925804181556418, −4.85461910560823371059418188057, −4.75110716284529228567779192130, −4.46061011725868076362322249460, −4.07258768172101631531743012042, −3.32321065157573113619754379714, −2.99871168567203333399958356231, −2.58048010489953929843379145581, −2.33602292609139606580468359735, −1.39131758895307238553349666187, −1.20526511451881696966041133531, −0.50083758549259891800268889200, 0.50083758549259891800268889200, 1.20526511451881696966041133531, 1.39131758895307238553349666187, 2.33602292609139606580468359735, 2.58048010489953929843379145581, 2.99871168567203333399958356231, 3.32321065157573113619754379714, 4.07258768172101631531743012042, 4.46061011725868076362322249460, 4.75110716284529228567779192130, 4.85461910560823371059418188057, 5.50892827791143925804181556418, 5.80683555839617801561362708889, 6.28592909289214005307878865614, 6.57003800194958448515494257296, 7.08490668393334585825389299390, 7.09105923397359768950600885078, 7.72408032474548797665328768976, 8.039410678429660950698247707711, 8.364146612788316031959783260767

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