Properties

Label 8-177e4-1.1-c3e4-0-0
Degree $8$
Conductor $981506241$
Sign $1$
Analytic cond. $11894.8$
Root an. cond. $3.23161$
Motivic weight $3$
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  − 7·3-s − 32·4-s − 58·7-s + 27·9-s + 224·12-s + 640·16-s − 238·19-s + 406·21-s − 191·25-s − 224·27-s + 1.85e3·28-s − 864·36-s − 4.48e3·48-s + 1.52e3·49-s + 1.66e3·57-s − 1.56e3·63-s − 1.02e4·64-s + 1.33e3·75-s + 7.61e3·76-s + 1.67e3·79-s + 1.56e3·81-s − 1.29e4·84-s + 6.11e3·100-s + 7.16e3·108-s − 3.71e4·112-s − 5.32e3·121-s + 127-s + ⋯
L(s)  = 1  − 1.34·3-s − 4·4-s − 3.13·7-s + 9-s + 5.38·12-s + 10·16-s − 2.87·19-s + 4.21·21-s − 1.52·25-s − 1.59·27-s + 12.5·28-s − 4·36-s − 13.4·48-s + 4.45·49-s + 3.87·57-s − 3.13·63-s − 20·64-s + 2.05·75-s + 11.4·76-s + 2.37·79-s + 2.15·81-s − 16.8·84-s + 6.11·100-s + 6.38·108-s − 31.3·112-s − 4·121-s + 0.000698·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 59^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 59^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 59^{4}\)
Sign: $1$
Analytic conductor: \(11894.8\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{177} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 59^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.0004170685410\)
\(L(\frac12)\) \(\approx\) \(0.0004170685410\)
\(L(\frac{5}{2})\) 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)$
bad3$C_2^2$ \( 1 + 7 T + 22 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
good2$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
5$C_2^2$$\times$$C_2^2$ \( ( 1 - 21 T + 316 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4} )( 1 + 21 T + 316 T^{2} + 21 p^{3} T^{3} + p^{6} T^{4} ) \)
7$C_2^2$ \( ( 1 + 29 T + 498 T^{2} + 29 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
11$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
13$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
17$C_2$ \( ( 1 - 126 T + p^{3} T^{2} )^{2}( 1 + 126 T + p^{3} T^{2} )^{2} \)
19$C_2^2$ \( ( 1 + 119 T + 7302 T^{2} + 119 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
23$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
29$C_2^2$$\times$$C_2^2$ \( ( 1 - 159 T + 892 T^{2} - 159 p^{3} T^{3} + p^{6} T^{4} )( 1 + 159 T + 892 T^{2} + 159 p^{3} T^{3} + p^{6} T^{4} ) \)
31$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
37$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
41$C_2^2$$\times$$C_2^2$ \( ( 1 - 525 T + 206704 T^{2} - 525 p^{3} T^{3} + p^{6} T^{4} )( 1 + 525 T + 206704 T^{2} + 525 p^{3} T^{3} + p^{6} T^{4} ) \)
43$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
47$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
53$C_2^2$$\times$$C_2^2$ \( ( 1 - 327 T - 41948 T^{2} - 327 p^{3} T^{3} + p^{6} T^{4} )( 1 + 327 T - 41948 T^{2} + 327 p^{3} T^{3} + p^{6} T^{4} ) \)
61$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
67$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
71$C_2$ \( ( 1 - 180 T + p^{3} T^{2} )^{2}( 1 + 180 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 835 T + 204186 T^{2} - 835 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
89$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
97$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
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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

−8.850156423830596864415932079742, −8.703130671531721083388515498000, −8.567013926483412586395556568972, −7.941712549700444723198189741399, −7.88568581792547979547884420356, −7.59428558578787852750538549269, −7.16064898845484828680573886858, −6.40171858580779191946942758084, −6.38726489432904388152097064027, −6.29291062911492051506644527401, −5.98382194852621079607289776790, −5.69731312289455907248978754614, −5.08288369691708611030722406455, −5.04493462268975431818553505039, −4.96808870443704163107983524371, −4.03510180718320369584739869492, −3.94397026756928190544901243350, −3.93232725477115576020955822793, −3.83200236561842820092989515820, −3.10488000042362624741191071477, −2.70309303190026504760401059898, −1.76306720346853333465005933389, −0.914810407908609963011206365389, −0.34200826286994347660510959214, −0.01938114876531591030430636149, 0.01938114876531591030430636149, 0.34200826286994347660510959214, 0.914810407908609963011206365389, 1.76306720346853333465005933389, 2.70309303190026504760401059898, 3.10488000042362624741191071477, 3.83200236561842820092989515820, 3.93232725477115576020955822793, 3.94397026756928190544901243350, 4.03510180718320369584739869492, 4.96808870443704163107983524371, 5.04493462268975431818553505039, 5.08288369691708611030722406455, 5.69731312289455907248978754614, 5.98382194852621079607289776790, 6.29291062911492051506644527401, 6.38726489432904388152097064027, 6.40171858580779191946942758084, 7.16064898845484828680573886858, 7.59428558578787852750538549269, 7.88568581792547979547884420356, 7.941712549700444723198189741399, 8.567013926483412586395556568972, 8.703130671531721083388515498000, 8.850156423830596864415932079742

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