Properties

Label 8-24e8-1.1-c3e4-0-6
Degree $8$
Conductor $110075314176$
Sign $1$
Analytic cond. $1.33399\times 10^{6}$
Root an. cond. $5.82967$
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  − 112·13-s + 464·25-s + 472·37-s + 988·49-s + 3.20e3·61-s + 1.98e3·73-s − 608·97-s − 2.51e3·109-s + 1.58e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 948·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2.38·13-s + 3.71·25-s + 2.09·37-s + 2.88·49-s + 6.73·61-s + 3.18·73-s − 0.636·97-s − 2.20·109-s + 1.19·121-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 − 0.431·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.33399\times 10^{6}\)
Root analytic conductor: \(5.82967\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(6.269254816\)
\(L(\frac12)\) \(\approx\) \(6.269254816\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5$C_2^2$ \( ( 1 - 232 T^{2} + p^{6} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 494 T^{2} + p^{6} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 794 T^{2} + p^{6} T^{4} )^{2} \)
13$C_2$ \( ( 1 + 28 T + p^{3} T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 304 T^{2} + p^{6} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 1430 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 6770 T^{2} + p^{6} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 48760 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 44030 T^{2} + p^{6} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 118 T + p^{3} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 101392 T^{2} + p^{6} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 96806 T^{2} + p^{6} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 121246 T^{2} + p^{6} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 192296 T^{2} + p^{6} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 65158 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 802 T + p^{3} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 582326 T^{2} + p^{6} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 297646 T^{2} + p^{6} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 496 T + p^{3} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 37090 T^{2} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 1140118 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 145888 T^{2} + p^{6} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 152 T + p^{3} T^{2} )^{4} \)
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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

−7.33953762580072678259974669268, −6.98864575217361363301507445638, −6.77833020669439155643847094486, −6.72432041553633815278862798850, −6.67217094474857227546706290763, −6.00044155657912531042877038745, −5.73910834858140830603939690620, −5.66208556526977843033295479035, −5.18783667955187402106885672412, −4.99610105816719006792892864212, −4.89442318761702390588080985422, −4.78454513419308175339164434715, −4.31202197529836163423913117540, −3.86986861560337311822893437546, −3.81329092510763989940237366109, −3.57855259241734728104682702545, −2.95065269984226824160168528182, −2.53076028495545884383275074870, −2.52929890566494137400391555735, −2.44107683985357646018353591316, −2.08118291008681836483270935405, −1.17474000960670353241072062523, −1.04283674366511077957858056541, −0.64046574234793390875610385008, −0.45581837417731065504076968022, 0.45581837417731065504076968022, 0.64046574234793390875610385008, 1.04283674366511077957858056541, 1.17474000960670353241072062523, 2.08118291008681836483270935405, 2.44107683985357646018353591316, 2.52929890566494137400391555735, 2.53076028495545884383275074870, 2.95065269984226824160168528182, 3.57855259241734728104682702545, 3.81329092510763989940237366109, 3.86986861560337311822893437546, 4.31202197529836163423913117540, 4.78454513419308175339164434715, 4.89442318761702390588080985422, 4.99610105816719006792892864212, 5.18783667955187402106885672412, 5.66208556526977843033295479035, 5.73910834858140830603939690620, 6.00044155657912531042877038745, 6.67217094474857227546706290763, 6.72432041553633815278862798850, 6.77833020669439155643847094486, 6.98864575217361363301507445638, 7.33953762580072678259974669268

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