Properties

Label 8-96e4-1.1-c2e4-0-1
Degree $8$
Conductor $84934656$
Sign $1$
Analytic cond. $46.8193$
Root an. cond. $1.61734$
Motivic weight $2$
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  − 6·9-s − 24·13-s + 84·25-s − 152·37-s + 20·49-s − 88·61-s − 120·73-s − 45·81-s + 360·97-s + 552·109-s + 144·117-s + 52·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 316·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2/3·9-s − 1.84·13-s + 3.35·25-s − 4.10·37-s + 0.408·49-s − 1.44·61-s − 1.64·73-s − 5/9·81-s + 3.71·97-s + 5.06·109-s + 1.23·117-s + 0.429·121-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 − 1.86·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(84934656\)    =    \(2^{20} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(46.8193\)
Root analytic conductor: \(1.61734\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{96} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 84934656,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.250694731\)
\(L(\frac12)\) \(\approx\) \(1.250694731\)
\(L(2)\) 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 \( 1 \)
3$C_2^2$ \( 1 + 2 p T^{2} + p^{4} T^{4} \)
good5$C_2^2$ \( ( 1 - 42 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 10 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 26 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2$ \( ( 1 + 6 T + p^{2} T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 66 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 614 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 194 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 714 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 950 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 38 T + p^{2} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 3330 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 3590 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 962 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 5418 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 6746 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 22 T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 4090 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 9218 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 30 T + p^{2} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 11510 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 8378 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 15810 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 90 T + p^{2} T^{2} )^{4} \)
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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

−10.10689332168073412800586003237, −10.01452804500833491457565862427, −9.398825701952516204538508999135, −9.016843606384522937152605004977, −8.886105192576732900615833238228, −8.812947937059081909045973510623, −8.409685389483895331322001040769, −8.094474269897240955774009407728, −7.46247041668435401666140127100, −7.35533197862825724121992851566, −7.03515121430873168330430166416, −6.95910415703906066547271563213, −6.33359633684317239176575653036, −6.16888777077254843860524860576, −5.53115874669343422208814807699, −5.25710819859385270392919339650, −5.00158943917870760443133334421, −4.55049468591177440319928361268, −4.49778979338763824838428107830, −3.49732992569682746460584074973, −3.10409391563928110667489198519, −3.05925057455437964479519765788, −2.22366461339349790618826363007, −1.73105121822732431778926147941, −0.54318900059203704815187276873, 0.54318900059203704815187276873, 1.73105121822732431778926147941, 2.22366461339349790618826363007, 3.05925057455437964479519765788, 3.10409391563928110667489198519, 3.49732992569682746460584074973, 4.49778979338763824838428107830, 4.55049468591177440319928361268, 5.00158943917870760443133334421, 5.25710819859385270392919339650, 5.53115874669343422208814807699, 6.16888777077254843860524860576, 6.33359633684317239176575653036, 6.95910415703906066547271563213, 7.03515121430873168330430166416, 7.35533197862825724121992851566, 7.46247041668435401666140127100, 8.094474269897240955774009407728, 8.409685389483895331322001040769, 8.812947937059081909045973510623, 8.886105192576732900615833238228, 9.016843606384522937152605004977, 9.398825701952516204538508999135, 10.01452804500833491457565862427, 10.10689332168073412800586003237

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