Properties

Label 8-55e4-1.1-c11e4-0-0
Degree $8$
Conductor $9150625$
Sign $1$
Analytic cond. $3.18912\times 10^{6}$
Root an. cond. $6.50068$
Motivic weight $11$
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  − 134·3-s + 8.97e3·9-s − 8.38e6·16-s − 9.09e7·23-s − 8.71e7·25-s + 2.31e7·27-s + 7.50e8·37-s − 4.69e9·47-s + 1.12e9·48-s − 1.21e10·53-s + 2.73e10·67-s + 1.21e10·69-s + 1.12e11·71-s + 1.16e10·75-s + 5.64e10·81-s − 3.03e11·97-s − 3.85e11·103-s − 1.00e11·111-s + 1.18e11·113-s + 5.70e11·121-s + 127-s + 131-s + 137-s + 139-s + 6.28e11·141-s − 7.53e10·144-s + 149-s + ⋯
L(s)  = 1  − 0.318·3-s + 0.0506·9-s − 2·16-s − 2.94·23-s − 1.78·25-s + 0.310·27-s + 1.77·37-s − 2.98·47-s + 0.636·48-s − 3.99·53-s + 2.47·67-s + 0.938·69-s + 7.37·71-s + 0.568·75-s + 1.79·81-s − 3.58·97-s − 3.27·103-s − 0.566·111-s + 0.606·113-s + 2·121-s + 0.950·141-s − 0.101·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(9150625\)    =    \(5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(3.18912\times 10^{6}\)
Root analytic conductor: \(6.50068\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 9150625,\ (\ :11/2, 11/2, 11/2, 11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(0.09471668032\)
\(L(\frac12)\) \(\approx\) \(0.09471668032\)
\(L(\frac{13}{2})\) 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)$
bad5$C_2^2$ \( 1 + 87113399 T^{2} + p^{22} T^{4} \)
11$C_2$ \( ( 1 - p^{11} T^{2} )^{2} \)
good2$C_2$ \( ( 1 - p^{6} T + p^{11} T^{2} )^{2}( 1 + p^{6} T + p^{11} T^{2} )^{2} \)
3$C_2$$\times$$C_2^2$ \( ( 1 + 67 T + p^{11} T^{2} )^{2}( 1 - 349805 T^{2} + p^{22} T^{4} ) \)
7$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
19$C_2$ \( ( 1 + p^{11} T^{2} )^{4} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + 45497817 T + p^{11} T^{2} )^{2}( 1 + 164431835937635 T^{2} + p^{22} T^{4} ) \)
29$C_2$ \( ( 1 + p^{11} T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 34677411476888363 T^{2} + p^{22} T^{4} )^{2} \)
37$C_2$$\times$$C_2^2$ \( ( 1 - 375115349 T + p^{11} T^{2} )^{2}( 1 - 215123718503529025 T^{2} + p^{22} T^{4} ) \)
41$C_2$ \( ( 1 - p^{11} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
47$C_2$$\times$$C_2^2$ \( ( 1 + 2346355716 T + p^{11} T^{2} )^{2}( 1 + 561066715837848050 T^{2} + p^{22} T^{4} ) \)
53$C_2$$\times$$C_2^2$ \( ( 1 + 6084080382 T + p^{11} T^{2} )^{2}( 1 + 18477962235892882730 T^{2} + p^{22} T^{4} ) \)
59$C_2$ \( ( 1 - 8051651835 T + p^{11} T^{2} )^{2}( 1 + 8051651835 T + p^{11} T^{2} )^{2} \)
61$C_2$ \( ( 1 - p^{11} T^{2} )^{4} \)
67$C_2$$\times$$C_2^2$ \( ( 1 - 13691120599 T + p^{11} T^{2} )^{2}( 1 - 56813482553573915365 T^{2} + p^{22} T^{4} ) \)
71$C_2$ \( ( 1 - 28030204947 T + p^{11} T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
79$C_2$ \( ( 1 + p^{11} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + \)\(31\!\cdots\!03\)\( T^{2} + p^{22} T^{4} )^{2} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + 151692012401 T + p^{11} T^{2} )^{2}( 1 + \)\(87\!\cdots\!95\)\( T^{2} + p^{22} T^{4} ) \)
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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

−9.248181273885664876480405203681, −8.235121432924754790682918095939, −8.186709359846838528671513681509, −8.179161693881651244108978101164, −7.911049902828729328158398245983, −7.46400001987461544002557880632, −6.71132359737950997119984054955, −6.55660237007009734022276124597, −6.45889520917173078553889518044, −6.24712796872425606160336184888, −5.65033100835122432034843724886, −5.22349556148705877719182898009, −4.89553866603318527249828782188, −4.75736064032734649197741755048, −3.90902161549296746510400242260, −3.89556849140984833603538790868, −3.85583076330173885939470055297, −2.93746106129601929557968623087, −2.65519722941006501843143437396, −2.00778313443388472897475034745, −1.95938153843491224742609314742, −1.73242463817929538387548686097, −0.935189212914407472360669271497, −0.49571676221562752689309590627, −0.05836961307251049038221993010, 0.05836961307251049038221993010, 0.49571676221562752689309590627, 0.935189212914407472360669271497, 1.73242463817929538387548686097, 1.95938153843491224742609314742, 2.00778313443388472897475034745, 2.65519722941006501843143437396, 2.93746106129601929557968623087, 3.85583076330173885939470055297, 3.89556849140984833603538790868, 3.90902161549296746510400242260, 4.75736064032734649197741755048, 4.89553866603318527249828782188, 5.22349556148705877719182898009, 5.65033100835122432034843724886, 6.24712796872425606160336184888, 6.45889520917173078553889518044, 6.55660237007009734022276124597, 6.71132359737950997119984054955, 7.46400001987461544002557880632, 7.911049902828729328158398245983, 8.179161693881651244108978101164, 8.186709359846838528671513681509, 8.235121432924754790682918095939, 9.248181273885664876480405203681

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