Properties

Label 8-960e4-1.1-c1e4-0-18
Degree $8$
Conductor $849346560000$
Sign $1$
Analytic cond. $3452.97$
Root an. cond. $2.76868$
Motivic weight $1$
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  + 6·9-s − 16·13-s − 2·25-s + 16·37-s + 4·49-s + 40·61-s − 8·73-s + 27·81-s + 40·97-s − 40·109-s − 96·117-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2·9-s − 4.43·13-s − 2/5·25-s + 2.63·37-s + 4/7·49-s + 5.12·61-s − 0.936·73-s + 3·81-s + 4.06·97-s − 3.83·109-s − 8.87·117-s − 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(3452.97\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{4} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.676076429\)
\(L(\frac12)\) \(\approx\) \(2.676076429\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
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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

−7.11497971684968336468316248350, −6.99383880967085274439688156584, −6.82580653174917820999052156720, −6.69837545135997129217770093922, −6.50209597120957223609352888581, −5.91290086632325466704977949110, −5.81992161310353618861476197595, −5.59029648855747847446993202119, −5.11488404002943330331590815223, −5.01015137904076881958405818697, −5.00500413134931241950736903349, −4.62727704328125151185581494693, −4.30505036211368661504363219478, −4.27254604993901083543535084823, −3.89865217245323185442279801352, −3.75193665852114912527313561294, −3.29956783434892709884503642743, −2.74519630418242092018695059058, −2.66395661926230098667515205010, −2.41587507820056681957513959122, −2.06121599581559811849482216811, −2.00009676375686446543975626198, −1.35594937719992004067689183599, −0.76714289168996611660210428909, −0.48937519566131699730116159891, 0.48937519566131699730116159891, 0.76714289168996611660210428909, 1.35594937719992004067689183599, 2.00009676375686446543975626198, 2.06121599581559811849482216811, 2.41587507820056681957513959122, 2.66395661926230098667515205010, 2.74519630418242092018695059058, 3.29956783434892709884503642743, 3.75193665852114912527313561294, 3.89865217245323185442279801352, 4.27254604993901083543535084823, 4.30505036211368661504363219478, 4.62727704328125151185581494693, 5.00500413134931241950736903349, 5.01015137904076881958405818697, 5.11488404002943330331590815223, 5.59029648855747847446993202119, 5.81992161310353618861476197595, 5.91290086632325466704977949110, 6.50209597120957223609352888581, 6.69837545135997129217770093922, 6.82580653174917820999052156720, 6.99383880967085274439688156584, 7.11497971684968336468316248350

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