Properties

Label 8-2e28-1.1-c3e4-0-0
Degree $8$
Conductor $268435456$
Sign $1$
Analytic cond. $3253.15$
Root an. cond. $2.74813$
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  + 28·9-s + 280·17-s − 140·25-s − 728·41-s − 348·49-s + 3.64e3·73-s − 870·81-s − 2.18e3·89-s − 1.96e3·97-s − 3.64e3·113-s + 1.40e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 7.84e3·153-s + 157-s + 163-s + 167-s + 8.14e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 1.03·9-s + 3.99·17-s − 1.11·25-s − 2.77·41-s − 1.01·49-s + 5.83·73-s − 1.19·81-s − 2.60·89-s − 2.05·97-s − 3.03·113-s + 1.05·121-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 + 4.14·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 3.70·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28}\)
Sign: $1$
Analytic conductor: \(3253.15\)
Root analytic conductor: \(2.74813\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(3.711962442\)
\(L(\frac12)\) \(\approx\) \(3.711962442\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good3$C_2^2$ \( ( 1 - 14 T^{2} + p^{6} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 + 14 p T^{2} + p^{6} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 174 T^{2} + p^{6} T^{4} )^{2} \)
11$C_2$ \( ( 1 - 58 T + p^{3} T^{2} )^{2}( 1 + 58 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( ( 1 - 4074 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2$ \( ( 1 - 70 T + p^{3} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 6958 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 754 T^{2} + p^{6} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 33098 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 39814 T^{2} + p^{6} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 182 T + p^{3} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 141374 T^{2} + p^{6} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 107294 T^{2} + p^{6} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 282074 T^{2} + p^{6} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 403998 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 399882 T^{2} + p^{6} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 552526 T^{2} + p^{6} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 703022 T^{2} + p^{6} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 910 T + p^{3} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 525278 T^{2} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 632814 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2$ \( ( 1 + 546 T + p^{3} T^{2} )^{4} \)
97$C_2$ \( ( 1 + 490 T + p^{3} T^{2} )^{4} \)
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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

−9.582775670188518468128407765530, −9.179098137008116716528552416196, −8.730562439661298931194605571401, −8.163832432087228483075589236271, −8.082688529071269249181523519771, −7.934803100557612782905712947515, −7.912089215880290909214454444923, −7.22580022671621646331627904796, −6.93904359422186065305981708579, −6.74195622579686793005980169405, −6.61216925073114069213018639094, −5.71317941436571232451105364763, −5.70107624936722437144644396843, −5.57790411167408496988752528083, −5.00896723109477689334745653313, −4.87596700180645501866596888215, −4.22586096781788846847025218758, −3.79679415022851530481763655912, −3.62644832231848533663856288426, −3.15706713652802246004115415058, −2.91788851574558275541648851809, −2.01542653512916390805880529757, −1.58060051165689216341964464967, −1.20462889427626375237706648641, −0.52240111283131630807705731875, 0.52240111283131630807705731875, 1.20462889427626375237706648641, 1.58060051165689216341964464967, 2.01542653512916390805880529757, 2.91788851574558275541648851809, 3.15706713652802246004115415058, 3.62644832231848533663856288426, 3.79679415022851530481763655912, 4.22586096781788846847025218758, 4.87596700180645501866596888215, 5.00896723109477689334745653313, 5.57790411167408496988752528083, 5.70107624936722437144644396843, 5.71317941436571232451105364763, 6.61216925073114069213018639094, 6.74195622579686793005980169405, 6.93904359422186065305981708579, 7.22580022671621646331627904796, 7.912089215880290909214454444923, 7.934803100557612782905712947515, 8.082688529071269249181523519771, 8.163832432087228483075589236271, 8.730562439661298931194605571401, 9.179098137008116716528552416196, 9.582775670188518468128407765530

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