Properties

Label 8-192e4-1.1-c3e4-0-4
Degree $8$
Conductor $1358954496$
Sign $1$
Analytic cond. $16469.0$
Root an. cond. $3.36576$
Motivic weight $3$
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  + 42·9-s + 104·13-s − 76·25-s − 824·37-s + 772·49-s − 1.11e3·61-s − 1.68e3·73-s + 1.03e3·81-s − 4.28e3·97-s − 1.43e3·109-s + 4.36e3·117-s − 3.59e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.02e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 14/9·9-s + 2.21·13-s − 0.607·25-s − 3.66·37-s + 2.25·49-s − 2.33·61-s − 2.70·73-s + 1.41·81-s − 4.48·97-s − 1.25·109-s + 3.45·117-s − 2.70·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.923·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(16469.0\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(2.028356504\)
\(L(\frac12)\) \(\approx\) \(2.028356504\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 - 14 p T^{2} + p^{6} T^{4} \)
good5$C_2^2$ \( ( 1 + 38 T^{2} + p^{6} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 386 T^{2} + p^{6} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 1798 T^{2} + p^{6} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 2 p T + p^{3} T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 5218 T^{2} + p^{6} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 2186 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 6770 T^{2} + p^{6} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 48490 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 58610 T^{2} + p^{6} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 206 T + p^{3} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 44530 T^{2} + p^{6} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 150266 T^{2} + p^{6} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 193822 T^{2} + p^{6} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 295162 T^{2} + p^{6} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 98854 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 278 T + p^{3} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 191062 T^{2} + p^{6} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 712366 T^{2} + p^{6} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 422 T + p^{3} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 539090 T^{2} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 1142710 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 1270546 T^{2} + p^{6} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 1070 T + p^{3} T^{2} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.667010718951912131482805044013, −8.461728742296701997767985863777, −8.286059964108047490627256026812, −7.898803733883064016553794049638, −7.51465416148392328939777908135, −7.23858673239299916003238330386, −7.02358482762530495570447157364, −6.99779084537303488304795442218, −6.28350044840629938410120434809, −6.26134310222814292622176743740, −6.05543776285219732014035139823, −5.46080414388818819171037104086, −5.29537595702398225095889412010, −5.09109086617813649083046122909, −4.48095945985781695147754493775, −4.10167408959360952729375887294, −3.93972410355055321193811451423, −3.81693371236585325364567291328, −3.23471156146629047639679896356, −2.91473677449186139773152097794, −2.37294450124460525331379994650, −1.57332790243082677000825871038, −1.50355090523781583354615996836, −1.26675236284137221341719895586, −0.27973072640983627071548291840, 0.27973072640983627071548291840, 1.26675236284137221341719895586, 1.50355090523781583354615996836, 1.57332790243082677000825871038, 2.37294450124460525331379994650, 2.91473677449186139773152097794, 3.23471156146629047639679896356, 3.81693371236585325364567291328, 3.93972410355055321193811451423, 4.10167408959360952729375887294, 4.48095945985781695147754493775, 5.09109086617813649083046122909, 5.29537595702398225095889412010, 5.46080414388818819171037104086, 6.05543776285219732014035139823, 6.26134310222814292622176743740, 6.28350044840629938410120434809, 6.99779084537303488304795442218, 7.02358482762530495570447157364, 7.23858673239299916003238330386, 7.51465416148392328939777908135, 7.898803733883064016553794049638, 8.286059964108047490627256026812, 8.461728742296701997767985863777, 8.667010718951912131482805044013

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