Properties

Label 8-3330e4-1.1-c1e4-0-9
Degree $8$
Conductor $1.230\times 10^{14}$
Sign $1$
Analytic cond. $499902.$
Root an. cond. $5.15656$
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  − 2·4-s + 20·11-s + 3·16-s + 20·19-s − 8·25-s + 8·31-s + 32·41-s − 40·44-s + 22·49-s − 24·59-s + 8·61-s − 4·64-s + 32·71-s − 40·76-s − 32·79-s − 28·89-s + 16·100-s − 24·101-s − 36·109-s + 210·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 4-s + 6.03·11-s + 3/4·16-s + 4.58·19-s − 8/5·25-s + 1.43·31-s + 4.99·41-s − 6.03·44-s + 22/7·49-s − 3.12·59-s + 1.02·61-s − 1/2·64-s + 3.79·71-s − 4.58·76-s − 3.60·79-s − 2.96·89-s + 8/5·100-s − 2.38·101-s − 3.44·109-s + 19.0·121-s − 1.43·124-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(499902.\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(13.90217721\)
\(L(\frac12)\) \(\approx\) \(13.90217721\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3 \( 1 \)
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
good7$D_4\times C_2$ \( 1 - 22 T^{2} + 211 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 - 10 T + 45 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 14 T^{2} + 315 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 14 T^{2} + 427 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 - 10 T + 55 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 16 T + 144 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 20 T^{2} + 2646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 + 16 T^{2} + 62 p T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 194 T^{2} + 14995 T^{4} - 194 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 12 T + 104 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 244 T^{2} + 23734 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 16 T + 204 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 226 T^{2} + 22627 T^{4} - 226 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 16 T + 204 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 118 T^{2} + 13731 T^{4} - 118 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 8 T^{2} - 4494 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.01905877937670264037145450154, −5.89415598897672982755498625803, −5.81929457276584053449617401901, −5.47052422745184447757811176644, −5.43894199230826200695674049974, −5.27848962296933944212535159378, −4.63516176387488827560160316612, −4.59050356222645476901411065216, −4.28432362210170583100084624115, −4.22543932792871929260577073227, −4.01876883823998604344942533598, −3.97723828511421997529963532265, −3.80906244822043070276087137758, −3.39963307805160160114423976814, −3.20150914562584287865537897661, −3.19773671114613222947669456200, −2.69457423616743438358077326095, −2.45364399517417283765646056628, −2.23581154928932571890023164183, −1.50116643417138729373587037472, −1.37348451999827008054575880785, −1.30100232767086296715843959866, −1.11730328185203863398007475186, −0.821658732579419575825030582974, −0.63063954992650479082756748250, 0.63063954992650479082756748250, 0.821658732579419575825030582974, 1.11730328185203863398007475186, 1.30100232767086296715843959866, 1.37348451999827008054575880785, 1.50116643417138729373587037472, 2.23581154928932571890023164183, 2.45364399517417283765646056628, 2.69457423616743438358077326095, 3.19773671114613222947669456200, 3.20150914562584287865537897661, 3.39963307805160160114423976814, 3.80906244822043070276087137758, 3.97723828511421997529963532265, 4.01876883823998604344942533598, 4.22543932792871929260577073227, 4.28432362210170583100084624115, 4.59050356222645476901411065216, 4.63516176387488827560160316612, 5.27848962296933944212535159378, 5.43894199230826200695674049974, 5.47052422745184447757811176644, 5.81929457276584053449617401901, 5.89415598897672982755498625803, 6.01905877937670264037145450154

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