Properties

Label 4-63e4-1.1-c0e2-0-8
Degree $4$
Conductor $15752961$
Sign $1$
Analytic cond. $3.92352$
Root an. cond. $1.40740$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 3·16-s − 25-s + 2·37-s + 2·43-s − 4·64-s + 4·67-s + 4·79-s + 2·100-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s − 4·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 2·4-s + 3·16-s − 25-s + 2·37-s + 2·43-s − 4·64-s + 4·67-s + 4·79-s + 2·100-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s − 4·172-s + 173-s + 179-s + 181-s + 191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15752961\)    =    \(3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(3.92352\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15752961,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9364221386\)
\(L(\frac12)\) \(\approx\) \(0.9364221386\)
\(L(1)\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2$ \( ( 1 + T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_2^2$ \( 1 - T^{2} + T^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2$ \( ( 1 - T + T^{2} )^{2} \)
41$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
43$C_2$ \( ( 1 - T + T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2^2$ \( 1 - T^{2} + T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_1$ \( ( 1 - T )^{4} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
79$C_1$ \( ( 1 - T )^{4} \)
83$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.730356349045030904179477848138, −8.477871022431795773556003663855, −8.122830317066194019739345565571, −7.912239764178354488690315617026, −7.43366841098501318227454996021, −7.21374550682274048826263045610, −6.31611146052426385825735333878, −6.30882740193674643537076218907, −5.63008439754334195154618621950, −5.57306651088991432208630119969, −4.88573270404298101204509525849, −4.77721636568213698627598579167, −4.10985503883110089462179773535, −4.09363921636143933977865234583, −3.41014954310708957042753152505, −3.29365954359094964092295553037, −2.24873927156594603547048436836, −2.17288818483861411014490213707, −0.917085110099472635406967331121, −0.791363440723084137400474316082, 0.791363440723084137400474316082, 0.917085110099472635406967331121, 2.17288818483861411014490213707, 2.24873927156594603547048436836, 3.29365954359094964092295553037, 3.41014954310708957042753152505, 4.09363921636143933977865234583, 4.10985503883110089462179773535, 4.77721636568213698627598579167, 4.88573270404298101204509525849, 5.57306651088991432208630119969, 5.63008439754334195154618621950, 6.30882740193674643537076218907, 6.31611146052426385825735333878, 7.21374550682274048826263045610, 7.43366841098501318227454996021, 7.912239764178354488690315617026, 8.122830317066194019739345565571, 8.477871022431795773556003663855, 8.730356349045030904179477848138

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