Properties

Label 8-50e4-1.1-c21e4-0-3
Degree $8$
Conductor $6250000$
Sign $1$
Analytic cond. $3.81299\times 10^{8}$
Root an. cond. $11.8211$
Motivic weight $21$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.09e6·4-s − 5.04e9·9-s + 4.37e10·11-s + 3.29e12·16-s + 1.04e14·19-s − 5.28e14·29-s + 4.25e15·31-s + 1.05e16·36-s + 4.42e17·41-s − 9.17e16·44-s + 1.17e18·49-s − 9.50e18·59-s + 9.49e18·61-s − 4.61e18·64-s + 4.40e18·71-s − 2.18e20·76-s + 8.47e19·79-s + 1.07e19·81-s − 5.11e20·89-s − 2.20e20·99-s + 1.77e21·101-s − 3.76e21·109-s + 1.10e21·116-s − 2.33e22·121-s − 8.91e21·124-s + ⋯
L(s)  = 1  − 4-s − 0.482·9-s + 0.508·11-s + 3/4·16-s + 3.89·19-s − 0.233·29-s + 0.931·31-s + 0.482·36-s + 5.14·41-s − 0.508·44-s + 2.09·49-s − 2.42·59-s + 1.70·61-s − 1/2·64-s + 0.160·71-s − 3.89·76-s + 1.00·79-s + 0.0986·81-s − 1.73·89-s − 0.245·99-s + 1.59·101-s − 1.52·109-s + 0.233·116-s − 3.15·121-s − 0.931·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s+21/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(6250000\)    =    \(2^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(3.81299\times 10^{8}\)
Root analytic conductor: \(11.8211\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 6250000,\ (\ :21/2, 21/2, 21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(4.755798872\)
\(L(\frac12)\) \(\approx\) \(4.755798872\)
\(L(\frac{23}{2})\) 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$C_2$ \( ( 1 + p^{20} T^{2} )^{2} \)
5 \( 1 \)
good3$D_4\times C_2$ \( 1 + 560321380 p^{2} T^{2} + 247884105734182 p^{10} T^{4} + 560321380 p^{44} T^{6} + p^{84} T^{8} \)
7$D_4\times C_2$ \( 1 - 23926196690353100 p^{2} T^{2} + \)\(33\!\cdots\!98\)\( p^{4} T^{4} - 23926196690353100 p^{44} T^{6} + p^{84} T^{8} \)
11$D_{4}$ \( ( 1 - 21869194224 T + \)\(12\!\cdots\!66\)\( T^{2} - 21869194224 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - \)\(76\!\cdots\!60\)\( T^{2} + \)\(15\!\cdots\!02\)\( p^{2} T^{4} - \)\(76\!\cdots\!60\)\( p^{42} T^{6} + p^{84} T^{8} \)
17$D_4\times C_2$ \( 1 - \)\(14\!\cdots\!20\)\( T^{2} + \)\(37\!\cdots\!02\)\( p^{2} T^{4} - \)\(14\!\cdots\!20\)\( p^{42} T^{6} + p^{84} T^{8} \)
19$D_{4}$ \( ( 1 - 52072861011560 T + \)\(91\!\cdots\!02\)\( p T^{2} - 52072861011560 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - \)\(69\!\cdots\!40\)\( T^{2} + \)\(40\!\cdots\!58\)\( T^{4} - \)\(69\!\cdots\!40\)\( p^{42} T^{6} + p^{84} T^{8} \)
29$D_{4}$ \( ( 1 + 264315081456060 T + \)\(28\!\cdots\!58\)\( T^{2} + 264315081456060 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 2126016880188664 T + \)\(36\!\cdots\!86\)\( T^{2} - 2126016880188664 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - \)\(32\!\cdots\!60\)\( T^{2} + \)\(40\!\cdots\!38\)\( T^{4} - \)\(32\!\cdots\!60\)\( p^{42} T^{6} + p^{84} T^{8} \)
41$D_{4}$ \( ( 1 - 221161031209870884 T + \)\(25\!\cdots\!46\)\( T^{2} - 221161031209870884 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(78\!\cdots\!00\)\( T^{2} + \)\(23\!\cdots\!98\)\( T^{4} - \)\(78\!\cdots\!00\)\( p^{42} T^{6} + p^{84} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(40\!\cdots\!80\)\( T^{2} + \)\(73\!\cdots\!18\)\( T^{4} - \)\(40\!\cdots\!80\)\( p^{42} T^{6} + p^{84} T^{8} \)
53$D_4\times C_2$ \( 1 - \)\(13\!\cdots\!80\)\( T^{2} + \)\(41\!\cdots\!18\)\( T^{4} - \)\(13\!\cdots\!80\)\( p^{42} T^{6} + p^{84} T^{8} \)
59$D_{4}$ \( ( 1 + 4752208294289856120 T + \)\(41\!\cdots\!02\)\( p T^{2} + 4752208294289856120 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 4748119038295687324 T + \)\(61\!\cdots\!66\)\( T^{2} - 4748119038295687324 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(62\!\cdots\!20\)\( T^{2} + \)\(19\!\cdots\!78\)\( T^{4} - \)\(62\!\cdots\!20\)\( p^{42} T^{6} + p^{84} T^{8} \)
71$D_{4}$ \( ( 1 - 2201035283486261544 T + \)\(49\!\cdots\!26\)\( T^{2} - 2201035283486261544 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - \)\(37\!\cdots\!40\)\( T^{2} + \)\(71\!\cdots\!58\)\( T^{4} - \)\(37\!\cdots\!40\)\( p^{42} T^{6} + p^{84} T^{8} \)
79$D_{4}$ \( ( 1 - 42382706176352134640 T + \)\(93\!\cdots\!58\)\( T^{2} - 42382706176352134640 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + \)\(12\!\cdots\!80\)\( T^{2} + \)\(79\!\cdots\!78\)\( T^{4} + \)\(12\!\cdots\!80\)\( p^{42} T^{6} + p^{84} T^{8} \)
89$D_{4}$ \( ( 1 + \)\(25\!\cdots\!80\)\( T + \)\(18\!\cdots\!78\)\( T^{2} + \)\(25\!\cdots\!80\)\( p^{21} T^{3} + p^{42} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(74\!\cdots\!80\)\( T^{2} + \)\(54\!\cdots\!18\)\( T^{4} - \)\(74\!\cdots\!80\)\( p^{42} T^{6} + p^{84} T^{8} \)
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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

−7.48962168875282752912847986961, −7.47929051549521051325634424864, −7.34273953035467541447688631624, −6.51233818872039417175017756994, −6.21421893882950211177726046440, −6.06874814747995062398886215064, −5.76405864462371495236240344508, −5.36724287778814482388957857152, −5.16567123559486270438894205788, −4.94809238988077000184099397883, −4.64248494486729775283893315171, −4.03517432018293647911478264056, −3.97708691260810712916228304919, −3.74607115944084311093802291181, −3.39887912351226729155351268830, −2.97682548156345517873023580680, −2.52670115372152460500114173734, −2.52576236325802506291892058353, −2.39036132116305903645881987980, −1.33461949110581972850553900841, −1.24326415165762654494283528632, −1.09694570829464008847895853171, −1.00831111670991470962563323825, −0.46893672753631565036018035023, −0.25557325170590954961116539963, 0.25557325170590954961116539963, 0.46893672753631565036018035023, 1.00831111670991470962563323825, 1.09694570829464008847895853171, 1.24326415165762654494283528632, 1.33461949110581972850553900841, 2.39036132116305903645881987980, 2.52576236325802506291892058353, 2.52670115372152460500114173734, 2.97682548156345517873023580680, 3.39887912351226729155351268830, 3.74607115944084311093802291181, 3.97708691260810712916228304919, 4.03517432018293647911478264056, 4.64248494486729775283893315171, 4.94809238988077000184099397883, 5.16567123559486270438894205788, 5.36724287778814482388957857152, 5.76405864462371495236240344508, 6.06874814747995062398886215064, 6.21421893882950211177726046440, 6.51233818872039417175017756994, 7.34273953035467541447688631624, 7.47929051549521051325634424864, 7.48962168875282752912847986961

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