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Properties

Label 8-5904e4-1.1-c1e4-0-2
Degree $8$
Conductor $1.215\times 10^{15}$
Sign $1$
Analytic cond. $4.93962\times 10^{6}$
Root an. cond. $6.86612$
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  − 4·5-s + 4·11-s + 4·17-s − 6·19-s − 12·23-s + 4·25-s + 4·29-s + 8·31-s + 16·37-s + 4·41-s − 4·43-s − 6·47-s − 6·49-s + 16·53-s − 16·55-s + 12·59-s + 24·61-s − 28·67-s − 2·71-s + 8·73-s + 18·79-s − 12·83-s − 16·85-s − 4·89-s + 24·95-s + 16·97-s + 12·101-s + ⋯
L(s)  = 1  − 1.78·5-s + 1.20·11-s + 0.970·17-s − 1.37·19-s − 2.50·23-s + 4/5·25-s + 0.742·29-s + 1.43·31-s + 2.63·37-s + 0.624·41-s − 0.609·43-s − 0.875·47-s − 6/7·49-s + 2.19·53-s − 2.15·55-s + 1.56·59-s + 3.07·61-s − 3.42·67-s − 0.237·71-s + 0.936·73-s + 2.02·79-s − 1.31·83-s − 1.73·85-s − 0.423·89-s + 2.46·95-s + 1.62·97-s + 1.19·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 41^{4}\)
Sign: $1$
Analytic conductor: \(4.93962\times 10^{6}\)
Root analytic conductor: \(6.86612\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} \cdot 41^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.697670500\)
\(L(\frac12)\) \(\approx\) \(5.697670500\)
\(L(\frac{3}{2})\) 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 \( 1 \)
41$C_1$ \( ( 1 - T )^{4} \)
good5$C_2 \wr S_4$ \( 1 + 4 T + 12 T^{2} + 16 T^{3} + 34 T^{4} + 16 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 6 T^{2} - 26 T^{3} + 24 T^{4} - 26 p T^{5} + 6 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 4 T + 26 T^{2} - 114 T^{3} + 384 T^{4} - 114 p T^{5} + 26 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 12 T^{2} - 48 T^{3} + 118 T^{4} - 48 p T^{5} + 12 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 4 T + 20 T^{2} - 124 T^{3} + 534 T^{4} - 124 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 6 T + 62 T^{2} + 208 T^{3} + 1448 T^{4} + 208 p T^{5} + 62 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 12 T + 108 T^{2} + 700 T^{3} + 3718 T^{4} + 700 p T^{5} + 108 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 4 T + 76 T^{2} - 12 p T^{3} + 2870 T^{4} - 12 p^{2} T^{5} + 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 8 T + 92 T^{2} - 712 T^{3} + 3846 T^{4} - 712 p T^{5} + 92 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 16 T + 212 T^{2} - 1740 T^{3} + 12626 T^{4} - 1740 p T^{5} + 212 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 4 T + 124 T^{2} + 244 T^{3} + 6678 T^{4} + 244 p T^{5} + 124 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 6 T + 126 T^{2} + 640 T^{3} + 8608 T^{4} + 640 p T^{5} + 126 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 16 T + 4 p T^{2} - 1824 T^{3} + 15558 T^{4} - 1824 p T^{5} + 4 p^{3} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 12 T + 252 T^{2} - 1996 T^{3} + 22582 T^{4} - 1996 p T^{5} + 252 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 24 T + 420 T^{2} - 4824 T^{3} + 44086 T^{4} - 4824 p T^{5} + 420 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 28 T + 538 T^{2} + 6638 T^{3} + 64208 T^{4} + 6638 p T^{5} + 538 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 2 T + 98 T^{2} - 268 T^{3} + 48 p T^{4} - 268 p T^{5} + 98 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 8 T + 212 T^{2} - 1060 T^{3} + 19890 T^{4} - 1060 p T^{5} + 212 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 18 T + 366 T^{2} - 4224 T^{3} + 45328 T^{4} - 4224 p T^{5} + 366 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 12 T + 252 T^{2} + 1644 T^{3} + 24598 T^{4} + 1644 p T^{5} + 252 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 4 T + 228 T^{2} + 796 T^{3} + 326 p T^{4} + 796 p T^{5} + 228 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 16 T + 268 T^{2} - 3376 T^{3} + 38118 T^{4} - 3376 p T^{5} + 268 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.77195423997759468603888520024, −5.44969687674583918515982014402, −5.36153156217550509457871893471, −5.00868902762684040859194069675, −4.98410444334466843117895652476, −4.50895584794149350389835443094, −4.40540119986193916440863419063, −4.34350763802441974277763010152, −4.14658366344712037910894585903, −4.00517122115985018107532535940, −3.73865859262591701421672899731, −3.72926390305732914839239231965, −3.50836083938274459560671994793, −3.03827650675780631779748770872, −2.94808806109812312328813059838, −2.72640250219277391987968470394, −2.64081095896958368169527420841, −1.98132677878411616179086020226, −1.93349708324418808998643079950, −1.87935567384746022545123085635, −1.63981692580207673794693109741, −0.838156293796803918705787422541, −0.794603752896167368683606535088, −0.53348722595245622583465642529, −0.53204922092879240220028234775, 0.53204922092879240220028234775, 0.53348722595245622583465642529, 0.794603752896167368683606535088, 0.838156293796803918705787422541, 1.63981692580207673794693109741, 1.87935567384746022545123085635, 1.93349708324418808998643079950, 1.98132677878411616179086020226, 2.64081095896958368169527420841, 2.72640250219277391987968470394, 2.94808806109812312328813059838, 3.03827650675780631779748770872, 3.50836083938274459560671994793, 3.72926390305732914839239231965, 3.73865859262591701421672899731, 4.00517122115985018107532535940, 4.14658366344712037910894585903, 4.34350763802441974277763010152, 4.40540119986193916440863419063, 4.50895584794149350389835443094, 4.98410444334466843117895652476, 5.00868902762684040859194069675, 5.36153156217550509457871893471, 5.44969687674583918515982014402, 5.77195423997759468603888520024

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