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Properties

Label 8-210e4-1.1-c3e4-0-2
Degree $8$
Conductor $1944810000$
Sign $1$
Analytic cond. $23569.0$
Root an. cond. $3.52000$
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  − 4·2-s + 6·3-s + 4·4-s − 10·5-s − 24·6-s − 6·7-s + 16·8-s + 9·9-s + 40·10-s + 20·11-s + 24·12-s + 84·13-s + 24·14-s − 60·15-s − 64·16-s − 76·17-s − 36·18-s − 90·19-s − 40·20-s − 36·21-s − 80·22-s − 44·23-s + 96·24-s + 25·25-s − 336·26-s − 54·27-s − 24·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s − 1.63·6-s − 0.323·7-s + 0.707·8-s + 1/3·9-s + 1.26·10-s + 0.548·11-s + 0.577·12-s + 1.79·13-s + 0.458·14-s − 1.03·15-s − 16-s − 1.08·17-s − 0.471·18-s − 1.08·19-s − 0.447·20-s − 0.374·21-s − 0.775·22-s − 0.398·23-s + 0.816·24-s + 1/5·25-s − 2.53·26-s − 0.384·27-s − 0.161·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(23569.0\)
Root analytic conductor: \(3.52000\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.2617762733\)
\(L(\frac12)\) \(\approx\) \(0.2617762733\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 + p T + p^{2} T^{2} )^{2} \)
3$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T + p^{2} T^{2} )^{2} \)
7$C_2^2$ \( 1 + 6 T - 65 p T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
good11$D_4\times C_2$ \( 1 - 20 T - 1212 T^{2} + 21000 T^{3} + 294583 T^{4} + 21000 p^{3} T^{5} - 1212 p^{6} T^{6} - 20 p^{9} T^{7} + p^{12} T^{8} \)
13$D_{4}$ \( ( 1 - 42 T + 4789 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 76 T + 72 T^{2} - 313272 T^{3} - 19979441 T^{4} - 313272 p^{3} T^{5} + 72 p^{6} T^{6} + 76 p^{9} T^{7} + p^{12} T^{8} \)
19$D_4\times C_2$ \( 1 + 90 T - 7459 T^{2} + 165690 T^{3} + 139478700 T^{4} + 165690 p^{3} T^{5} - 7459 p^{6} T^{6} + 90 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 + 44 T - 17316 T^{2} - 223608 T^{3} + 199048303 T^{4} - 223608 p^{3} T^{5} - 17316 p^{6} T^{6} + 44 p^{9} T^{7} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 + 160 T + 54028 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 2 p T - 53755 T^{2} - 3966 p T^{3} + 2315624516 T^{4} - 3966 p^{4} T^{5} - 53755 p^{6} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 - 358 T + 39023 T^{2} + 4355070 T^{3} - 1111499620 T^{4} + 4355070 p^{3} T^{5} + 39023 p^{6} T^{6} - 358 p^{9} T^{7} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 - 36 T + 2552 p T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 134 T + 129969 T^{2} + 134 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 684 T + 202862 T^{2} + 39226032 T^{3} + 10868391219 T^{4} + 39226032 p^{3} T^{5} + 202862 p^{6} T^{6} + 684 p^{9} T^{7} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 + 16 T - 182562 T^{2} - 1838976 T^{3} + 11219947483 T^{4} - 1838976 p^{3} T^{5} - 182562 p^{6} T^{6} + 16 p^{9} T^{7} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 + 552 T - 171880 T^{2} + 36335952 T^{3} + 122371972599 T^{4} + 36335952 p^{3} T^{5} - 171880 p^{6} T^{6} + 552 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 + 1312 T + 851950 T^{2} + 545046784 T^{3} + 313613170411 T^{4} + 545046784 p^{3} T^{5} + 851950 p^{6} T^{6} + 1312 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 + 194 T - 495973 T^{2} - 13175898 T^{3} + 182960666444 T^{4} - 13175898 p^{3} T^{5} - 495973 p^{6} T^{6} + 194 p^{9} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 - 380 T - 111452 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 198 T - 743065 T^{2} + 838530 T^{3} + 445359539604 T^{4} + 838530 p^{3} T^{5} - 743065 p^{6} T^{6} + 198 p^{9} T^{7} + p^{12} T^{8} \)
79$D_4\times C_2$ \( 1 + 126 T + 293405 T^{2} - 159214482 T^{3} - 173892596844 T^{4} - 159214482 p^{3} T^{5} + 293405 p^{6} T^{6} + 126 p^{9} T^{7} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 - 864 T + 336184 T^{2} - 864 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 184 T - 250692 T^{2} + 207071760 T^{3} - 439881157121 T^{4} + 207071760 p^{3} T^{5} - 250692 p^{6} T^{6} - 184 p^{9} T^{7} + p^{12} T^{8} \)
97$D_{4}$ \( ( 1 - 568 T + 1864602 T^{2} - 568 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−8.570082723507532031684492234357, −8.350691317718285726194379990170, −8.104825385850043500498278470573, −7.957897932702868941925685915188, −7.64839958205491263128134531092, −7.42891375107035685702209861138, −7.20651636696616838248019539442, −6.60523565514622552144915367937, −6.49145874832749720850202151226, −6.18670402296274295462315414909, −6.08144227878196622903871373209, −5.56163101012144150681456424135, −5.07718651364249218533317240116, −4.75384908206564451705324524583, −4.31478978103139535836459832802, −4.05006607774036186080986562464, −3.78086682424845259523998542333, −3.62644184421679226934622506785, −3.10038216938883727089728242335, −2.75945874214324766402009819211, −2.15778279177898211385808547534, −1.65664408793833869433363714088, −1.58891008892998587267561310216, −0.74911446209733302330179214928, −0.15312367793849538662497046440, 0.15312367793849538662497046440, 0.74911446209733302330179214928, 1.58891008892998587267561310216, 1.65664408793833869433363714088, 2.15778279177898211385808547534, 2.75945874214324766402009819211, 3.10038216938883727089728242335, 3.62644184421679226934622506785, 3.78086682424845259523998542333, 4.05006607774036186080986562464, 4.31478978103139535836459832802, 4.75384908206564451705324524583, 5.07718651364249218533317240116, 5.56163101012144150681456424135, 6.08144227878196622903871373209, 6.18670402296274295462315414909, 6.49145874832749720850202151226, 6.60523565514622552144915367937, 7.20651636696616838248019539442, 7.42891375107035685702209861138, 7.64839958205491263128134531092, 7.957897932702868941925685915188, 8.104825385850043500498278470573, 8.350691317718285726194379990170, 8.570082723507532031684492234357

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