Properties

Label 4-4032e2-1.1-c1e2-0-19
Degree $4$
Conductor $16257024$
Sign $1$
Analytic cond. $1036.56$
Root an. cond. $5.67412$
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  + 4·7-s + 4·19-s − 2·25-s + 12·29-s + 16·31-s + 4·37-s + 9·49-s + 12·53-s − 12·59-s + 12·83-s − 8·103-s − 28·109-s + 36·113-s + 10·121-s + 127-s + 131-s + 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s − 8·175-s + ⋯
L(s)  = 1  + 1.51·7-s + 0.917·19-s − 2/5·25-s + 2.22·29-s + 2.87·31-s + 0.657·37-s + 9/7·49-s + 1.64·53-s − 1.56·59-s + 1.31·83-s − 0.788·103-s − 2.68·109-s + 3.38·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.07·169-s + 0.0760·173-s − 0.604·175-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16257024\)    =    \(2^{12} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1036.56\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16257024,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.599326145\)
\(L(\frac12)\) \(\approx\) \(4.599326145\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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

−8.704333491648407150142376325945, −8.311492199513727407679115050624, −7.80706935359832165938737593226, −7.74632956708739299467140651858, −7.26804388829429283681088163736, −6.64929427669292913537578403352, −6.57071922797863295308727580018, −5.95299128062302412355606174340, −5.75640528600867934707596118030, −5.06081422891627967285505970636, −4.87871780078869952405542984790, −4.53999919407921005440800133066, −4.24952697708590079815601943601, −3.65368054408025903311939283254, −3.09548094053656136289574167579, −2.56074067948233261186936320156, −2.42795451632466793796997244096, −1.56681753392349750065218013904, −1.10721803633979057332405363837, −0.73005477831156902975680407883, 0.73005477831156902975680407883, 1.10721803633979057332405363837, 1.56681753392349750065218013904, 2.42795451632466793796997244096, 2.56074067948233261186936320156, 3.09548094053656136289574167579, 3.65368054408025903311939283254, 4.24952697708590079815601943601, 4.53999919407921005440800133066, 4.87871780078869952405542984790, 5.06081422891627967285505970636, 5.75640528600867934707596118030, 5.95299128062302412355606174340, 6.57071922797863295308727580018, 6.64929427669292913537578403352, 7.26804388829429283681088163736, 7.74632956708739299467140651858, 7.80706935359832165938737593226, 8.311492199513727407679115050624, 8.704333491648407150142376325945

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