Properties

Label 4-65088-1.1-c1e2-0-0
Degree $4$
Conductor $65088$
Sign $1$
Analytic cond. $4.15006$
Root an. cond. $1.42729$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·4-s + 9-s + 4·11-s − 4·12-s + 4·16-s + 2·19-s + 6·25-s − 4·27-s + 8·33-s − 2·36-s + 8·41-s − 14·43-s − 8·44-s + 8·48-s + 2·49-s + 4·57-s + 10·59-s − 8·64-s + 2·67-s + 20·73-s + 12·75-s − 4·76-s − 11·81-s − 20·89-s + 12·97-s + 4·99-s + ⋯
L(s)  = 1  + 1.15·3-s − 4-s + 1/3·9-s + 1.20·11-s − 1.15·12-s + 16-s + 0.458·19-s + 6/5·25-s − 0.769·27-s + 1.39·33-s − 1/3·36-s + 1.24·41-s − 2.13·43-s − 1.20·44-s + 1.15·48-s + 2/7·49-s + 0.529·57-s + 1.30·59-s − 64-s + 0.244·67-s + 2.34·73-s + 1.38·75-s − 0.458·76-s − 1.22·81-s − 2.11·89-s + 1.21·97-s + 0.402·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65088 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65088 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(65088\)    =    \(2^{6} \cdot 3^{2} \cdot 113\)
Sign: $1$
Analytic conductor: \(4.15006\)
Root analytic conductor: \(1.42729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 65088,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.775513951\)
\(L(\frac12)\) \(\approx\) \(1.775513951\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
113$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \)
61$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−9.763115112618116926593583340462, −9.253244850303103632918268422399, −8.934378157250735909274858290334, −8.405625494704305756939963067305, −8.164755510833238577774451528619, −7.42587913865188688795426729358, −6.89971322470879843858479366659, −6.28522008363674035684659848112, −5.54148455756633746835211789311, −4.94951204053151806021768680507, −4.29236972126143639377325873081, −3.62849897297158305526116002325, −3.25972353252661926041149408314, −2.28649166833768613469376324136, −1.13863093393384090818968127410, 1.13863093393384090818968127410, 2.28649166833768613469376324136, 3.25972353252661926041149408314, 3.62849897297158305526116002325, 4.29236972126143639377325873081, 4.94951204053151806021768680507, 5.54148455756633746835211789311, 6.28522008363674035684659848112, 6.89971322470879843858479366659, 7.42587913865188688795426729358, 8.164755510833238577774451528619, 8.405625494704305756939963067305, 8.934378157250735909274858290334, 9.253244850303103632918268422399, 9.763115112618116926593583340462

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