Properties

Label 8-33e8-1.1-c2e4-0-9
Degree $8$
Conductor $1.406\times 10^{12}$
Sign $1$
Analytic cond. $775267.$
Root an. cond. $5.44730$
Motivic weight $2$
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  + 20·7-s + 68·13-s − 17·16-s − 24·19-s + 60·25-s − 56·31-s + 72·37-s + 12·43-s + 84·49-s + 24·61-s − 52·67-s + 128·73-s + 136·79-s + 1.36e3·91-s + 176·97-s − 148·103-s + 20·109-s − 340·112-s + 127-s + 131-s − 480·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 20/7·7-s + 5.23·13-s − 1.06·16-s − 1.26·19-s + 12/5·25-s − 1.80·31-s + 1.94·37-s + 0.279·43-s + 12/7·49-s + 0.393·61-s − 0.776·67-s + 1.75·73-s + 1.72·79-s + 14.9·91-s + 1.81·97-s − 1.43·103-s + 0.183·109-s − 3.03·112-s + 0.00787·127-s + 0.00763·131-s − 3.60·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(775267.\)
Root analytic conductor: \(5.44730\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1089} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 11^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(16.77572696\)
\(L(\frac12)\) \(\approx\) \(16.77572696\)
\(L(2)\) 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 \( 1 \)
11 \( 1 \)
good2$C_2^3$ \( 1 + 17 T^{4} + p^{8} T^{8} \)
5$D_4\times C_2$ \( 1 - 12 p T^{2} + 71 p^{2} T^{4} - 12 p^{5} T^{6} + p^{8} T^{8} \)
7$D_{4}$ \( ( 1 - 10 T + 108 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
13$D_{4}$ \( ( 1 - 34 T + 567 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 996 T^{2} + 411671 T^{4} - 996 p^{4} T^{6} + p^{8} T^{8} \)
19$D_{4}$ \( ( 1 + 12 T + 218 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 1320 T^{2} + 857042 T^{4} - 1320 p^{4} T^{6} + p^{8} T^{8} \)
29$D_4\times C_2$ \( 1 - 1324 T^{2} + 1822431 T^{4} - 1324 p^{4} T^{6} + p^{8} T^{8} \)
31$D_{4}$ \( ( 1 + 28 T + 618 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 36 T + 2687 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 1500 T^{2} + 3920807 T^{4} + 1500 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 - 6 T - 628 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 8196 T^{2} + 26498966 T^{4} - 8196 p^{4} T^{6} + p^{8} T^{8} \)
53$D_4\times C_2$ \( 1 - 10060 T^{2} + 40757727 T^{4} - 10060 p^{4} T^{6} + p^{8} T^{8} \)
59$C_2^2$ \( ( 1 - 2160 T^{2} + p^{4} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 12 T + 6518 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 26 T + 9132 T^{2} + 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 13120 T^{2} + 88016322 T^{4} - 13120 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 64 T + 11142 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 68 T + 12678 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 11140 T^{2} + 86575542 T^{4} - 11140 p^{4} T^{6} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 8260 T^{2} + 142190247 T^{4} - 8260 p^{4} T^{6} + p^{8} T^{8} \)
97$D_{4}$ \( ( 1 - 88 T + 17379 T^{2} - 88 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−6.64749642883988668053022054011, −6.54315366785966387002567814429, −6.39443442331013067554235241733, −6.34545364721885904960781462803, −5.95714051890403671309830757112, −5.87088183753429079739816207087, −5.35005389266830876238299897617, −5.19890265298148709711659441656, −5.07283945756648761354538654417, −4.89426609159045627654247269654, −4.41228157826053193129738418023, −4.37971546164438833898447109328, −3.95885043340886240044830897365, −3.88784482737672385903390651281, −3.71702711256395072775408195565, −3.37785383047930865819188910154, −2.99172221056120840119533005485, −2.66940524547278730847312609424, −2.34336952943031546766718937945, −1.86391409557136897611391869061, −1.61569073827847634773787381173, −1.45906789195658397660211159000, −1.30247469953930932776155304065, −0.73131308295227474162414381945, −0.61807720387855055469328090350, 0.61807720387855055469328090350, 0.73131308295227474162414381945, 1.30247469953930932776155304065, 1.45906789195658397660211159000, 1.61569073827847634773787381173, 1.86391409557136897611391869061, 2.34336952943031546766718937945, 2.66940524547278730847312609424, 2.99172221056120840119533005485, 3.37785383047930865819188910154, 3.71702711256395072775408195565, 3.88784482737672385903390651281, 3.95885043340886240044830897365, 4.37971546164438833898447109328, 4.41228157826053193129738418023, 4.89426609159045627654247269654, 5.07283945756648761354538654417, 5.19890265298148709711659441656, 5.35005389266830876238299897617, 5.87088183753429079739816207087, 5.95714051890403671309830757112, 6.34545364721885904960781462803, 6.39443442331013067554235241733, 6.54315366785966387002567814429, 6.64749642883988668053022054011

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