Properties

Label 4-20e4-1.1-c2e2-0-3
Degree $4$
Conductor $160000$
Sign $1$
Analytic cond. $118.792$
Root an. cond. $3.30139$
Motivic weight $2$
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  − 18·9-s − 84·29-s + 36·41-s − 98·49-s − 44·61-s + 243·81-s + 156·89-s − 396·101-s + 364·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 238·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 2·9-s − 2.89·29-s + 0.878·41-s − 2·49-s − 0.721·61-s + 3·81-s + 1.75·89-s − 3.92·101-s + 3.33·109-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.40·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(160000\)    =    \(2^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(118.792\)
Root analytic conductor: \(3.30139\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 160000,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8382358994\)
\(L(\frac12)\) \(\approx\) \(0.8382358994\)
\(L(2)\) 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 \( 1 \)
good3$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
7$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
13$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \)
17$C_2$ \( ( 1 - 16 T + p^{2} T^{2} )( 1 + 16 T + p^{2} T^{2} ) \)
19$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
23$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 42 T + p^{2} T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
37$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \)
41$C_2$ \( ( 1 - 18 T + p^{2} T^{2} )^{2} \)
43$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
47$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
53$C_2$ \( ( 1 - 56 T + p^{2} T^{2} )( 1 + 56 T + p^{2} T^{2} ) \)
59$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
61$C_2$ \( ( 1 + 22 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 - 96 T + p^{2} T^{2} )( 1 + 96 T + p^{2} T^{2} ) \)
79$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
83$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
89$C_2$ \( ( 1 - 78 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 144 T + p^{2} T^{2} )( 1 + 144 T + p^{2} T^{2} ) \)
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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

−11.31631067654155939511873078681, −10.98044389346249706190555840985, −10.57196886845406671507478917666, −9.736158068018111996660235196658, −9.447735115897835704928472720304, −9.012921444602325913956290208206, −8.603650460550464131015636588966, −7.957838143130396509679657485971, −7.77916735748619070642189723664, −7.09825128271227200704992251827, −6.47757093837180039150001681037, −5.80159966307983912919239194640, −5.73375891209066438669387765747, −5.06889606558167421876942803591, −4.41826656913265296923151493440, −3.53838678391230767577441094724, −3.24198012716832184671050455868, −2.42960754311593289345520877901, −1.75671052191762062067725685878, −0.38274336472573446682407863325, 0.38274336472573446682407863325, 1.75671052191762062067725685878, 2.42960754311593289345520877901, 3.24198012716832184671050455868, 3.53838678391230767577441094724, 4.41826656913265296923151493440, 5.06889606558167421876942803591, 5.73375891209066438669387765747, 5.80159966307983912919239194640, 6.47757093837180039150001681037, 7.09825128271227200704992251827, 7.77916735748619070642189723664, 7.957838143130396509679657485971, 8.603650460550464131015636588966, 9.012921444602325913956290208206, 9.447735115897835704928472720304, 9.736158068018111996660235196658, 10.57196886845406671507478917666, 10.98044389346249706190555840985, 11.31631067654155939511873078681

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