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Properties

Label 8-10e8-1.1-c17e4-0-1
Degree $8$
Conductor $100000000$
Sign $1$
Analytic cond. $1.12696\times 10^{9}$
Root an. cond. $13.5359$
Motivic weight $17$
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  − 1.90e8·9-s + 2.51e9·11-s − 2.02e11·19-s − 4.67e12·29-s + 5.57e11·31-s + 8.33e12·41-s + 3.65e14·49-s − 5.67e15·59-s − 1.35e14·61-s + 7.76e15·71-s + 2.99e16·79-s + 5.90e15·81-s − 1.03e17·89-s − 4.80e17·99-s + 4.15e17·101-s + 8.52e17·109-s + 2.94e18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.54e19·169-s + ⋯
L(s)  = 1  − 1.47·9-s + 3.54·11-s − 2.73·19-s − 1.73·29-s + 0.117·31-s + 0.162·41-s + 1.57·49-s − 5.02·59-s − 0.0902·61-s + 1.42·71-s + 2.22·79-s + 0.354·81-s − 2.79·89-s − 5.23·99-s + 3.81·101-s + 4.09·109-s + 5.83·121-s + 1.78·169-s + 4.03·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(18-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+17/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.12696\times 10^{9}\)
Root analytic conductor: \(13.5359\)
Motivic weight: \(17\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{8} ,\ ( \ : 17/2, 17/2, 17/2, 17/2 ),\ 1 )\)

Particular Values

\(L(9)\) \(\approx\) \(6.492773651\)
\(L(\frac12)\) \(\approx\) \(6.492773651\)
\(L(\frac{19}{2})\) 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$D_4\times C_2$ \( 1 + 21210484 p^{2} T^{2} + 517086796678 p^{10} T^{4} + 21210484 p^{36} T^{6} + p^{68} T^{8} \)
7$D_4\times C_2$ \( 1 - 365690701232540 T^{2} + \)\(26\!\cdots\!98\)\( p^{4} T^{4} - 365690701232540 p^{34} T^{6} + p^{68} T^{8} \)
11$D_{4}$ \( ( 1 - 114513480 p T + 7489675618956502 p^{2} T^{2} - 114513480 p^{18} T^{3} + p^{34} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 15448183398079958444 T^{2} + \)\(11\!\cdots\!98\)\( p^{2} T^{4} - 15448183398079958444 p^{34} T^{6} + p^{68} T^{8} \)
17$D_4\times C_2$ \( 1 - \)\(29\!\cdots\!16\)\( T^{2} + \)\(34\!\cdots\!22\)\( T^{4} - \)\(29\!\cdots\!16\)\( p^{34} T^{6} + p^{68} T^{8} \)
19$D_{4}$ \( ( 1 + 101133633832 T + \)\(10\!\cdots\!34\)\( T^{2} + 101133633832 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - \)\(41\!\cdots\!40\)\( T^{2} + \)\(81\!\cdots\!18\)\( T^{4} - \)\(41\!\cdots\!40\)\( p^{34} T^{6} + p^{68} T^{8} \)
29$D_{4}$ \( ( 1 + 2337155582652 T + \)\(94\!\cdots\!94\)\( T^{2} + 2337155582652 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 278836113472 T + \)\(39\!\cdots\!18\)\( T^{2} - 278836113472 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - \)\(14\!\cdots\!20\)\( T^{2} + \)\(89\!\cdots\!78\)\( T^{4} - \)\(14\!\cdots\!20\)\( p^{34} T^{6} + p^{68} T^{8} \)
41$D_{4}$ \( ( 1 - 4166592315732 T + \)\(39\!\cdots\!18\)\( T^{2} - 4166592315732 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(15\!\cdots\!72\)\( T^{2} + \)\(11\!\cdots\!94\)\( T^{4} - \)\(15\!\cdots\!72\)\( p^{34} T^{6} + p^{68} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(68\!\cdots\!36\)\( T^{2} + \)\(22\!\cdots\!62\)\( T^{4} - \)\(68\!\cdots\!36\)\( p^{34} T^{6} + p^{68} T^{8} \)
53$D_4\times C_2$ \( 1 - \)\(60\!\cdots\!00\)\( T^{2} + \)\(16\!\cdots\!38\)\( T^{4} - \)\(60\!\cdots\!00\)\( p^{34} T^{6} + p^{68} T^{8} \)
59$D_{4}$ \( ( 1 + 2835904197813624 T + \)\(45\!\cdots\!82\)\( T^{2} + 2835904197813624 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 67544034994436 T - \)\(59\!\cdots\!34\)\( T^{2} + 67544034994436 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(35\!\cdots\!40\)\( T^{2} + \)\(55\!\cdots\!58\)\( T^{4} - \)\(35\!\cdots\!40\)\( p^{34} T^{6} + p^{68} T^{8} \)
71$D_{4}$ \( ( 1 - 3882245493215376 T + \)\(21\!\cdots\!26\)\( T^{2} - 3882245493215376 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - \)\(10\!\cdots\!64\)\( T^{2} + \)\(74\!\cdots\!42\)\( T^{4} - \)\(10\!\cdots\!64\)\( p^{34} T^{6} + p^{68} T^{8} \)
79$D_{4}$ \( ( 1 - 14984271534065504 T + \)\(37\!\cdots\!22\)\( T^{2} - 14984271534065504 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(68\!\cdots\!60\)\( T^{2} + \)\(43\!\cdots\!58\)\( T^{4} - \)\(68\!\cdots\!60\)\( p^{34} T^{6} + p^{68} T^{8} \)
89$D_{4}$ \( ( 1 + 51909007958846388 T + \)\(34\!\cdots\!94\)\( T^{2} + 51909007958846388 p^{17} T^{3} + p^{34} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(15\!\cdots\!20\)\( T^{2} + \)\(12\!\cdots\!38\)\( T^{4} - \)\(15\!\cdots\!20\)\( p^{34} T^{6} + p^{68} T^{8} \)
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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

−7.23822097996682139365708649022, −6.54056764018200764871914811926, −6.47505389686663566200138963144, −6.27550314131725202949416493610, −6.16176306369123194852163769877, −5.95567641094433827072597958665, −5.49981856044616283459754417923, −5.21250865789050504051446045648, −4.73800971950281363580619028380, −4.39509658568080559027145696122, −4.39441399436452866475328970594, −3.94538332060460301494189569755, −3.77921728671010773986530814787, −3.40815855575790964132465106242, −3.35512050611919652285812369491, −2.85039601627359153755865456045, −2.41228360596707550064194140130, −2.28627902372677260033749357870, −1.79918991072663847908337506735, −1.72346000047148461470371913533, −1.38668036843394120299846721151, −1.20015576234579799170413352396, −0.55558768519913018421391145947, −0.39325998194105747370767651855, −0.37758116571661096443163484644, 0.37758116571661096443163484644, 0.39325998194105747370767651855, 0.55558768519913018421391145947, 1.20015576234579799170413352396, 1.38668036843394120299846721151, 1.72346000047148461470371913533, 1.79918991072663847908337506735, 2.28627902372677260033749357870, 2.41228360596707550064194140130, 2.85039601627359153755865456045, 3.35512050611919652285812369491, 3.40815855575790964132465106242, 3.77921728671010773986530814787, 3.94538332060460301494189569755, 4.39441399436452866475328970594, 4.39509658568080559027145696122, 4.73800971950281363580619028380, 5.21250865789050504051446045648, 5.49981856044616283459754417923, 5.95567641094433827072597958665, 6.16176306369123194852163769877, 6.27550314131725202949416493610, 6.47505389686663566200138963144, 6.54056764018200764871914811926, 7.23822097996682139365708649022

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