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Properties

Label 8-1904e4-1.1-c1e4-0-4
Degree $8$
Conductor $1.314\times 10^{13}$
Sign $1$
Analytic cond. $53428.8$
Root an. cond. $3.89916$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 5-s + 4·7-s + 2·9-s − 4·11-s − 8·13-s + 3·15-s + 4·17-s − 14·19-s − 12·21-s − 8·23-s − 4·25-s + 2·27-s + 4·29-s − 5·31-s + 12·33-s − 4·35-s − 4·37-s + 24·39-s − 7·41-s − 19·43-s − 2·45-s − 8·47-s + 10·49-s − 12·51-s + 5·53-s + 4·55-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.447·5-s + 1.51·7-s + 2/3·9-s − 1.20·11-s − 2.21·13-s + 0.774·15-s + 0.970·17-s − 3.21·19-s − 2.61·21-s − 1.66·23-s − 4/5·25-s + 0.384·27-s + 0.742·29-s − 0.898·31-s + 2.08·33-s − 0.676·35-s − 0.657·37-s + 3.84·39-s − 1.09·41-s − 2.89·43-s − 0.298·45-s − 1.16·47-s + 10/7·49-s − 1.68·51-s + 0.686·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 7^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(53428.8\)
Root analytic conductor: \(3.89916\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 7^{4} \cdot 17^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{4} \)
17$C_1$ \( ( 1 - T )^{4} \)
good3$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + p T + 7 T^{2} + 13 T^{3} + 28 T^{4} + 13 p T^{5} + 7 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + T + p T^{2} - 7 T^{3} + 4 T^{4} - 7 p T^{5} + p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_{4}$ \( ( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 116 T^{2} + 710 T^{3} + 3510 T^{4} + 710 p T^{5} + 116 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 72 T^{2} + 328 T^{3} + 1998 T^{4} + 328 p T^{5} + 72 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 48 T^{2} - 364 T^{3} + 1118 T^{4} - 364 p T^{5} + 48 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 39 T^{2} - 35 T^{3} + 96 T^{4} - 35 p T^{5} + 39 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 80 T^{2} + 140 T^{3} + 2878 T^{4} + 140 p T^{5} + 80 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 99 T^{2} + 625 T^{3} + 5720 T^{4} + 625 p T^{5} + 99 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 19 T + 237 T^{2} + 2183 T^{3} + 16508 T^{4} + 2183 p T^{5} + 237 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 76 T^{2} + 616 T^{3} + 5542 T^{4} + 616 p T^{5} + 76 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 111 T^{2} - 575 T^{3} + 6632 T^{4} - 575 p T^{5} + 111 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 192 T^{2} - 80 T^{3} + 15758 T^{4} - 80 p T^{5} + 192 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 23 T + 357 T^{2} + 3847 T^{3} + 34148 T^{4} + 3847 p T^{5} + 357 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 239 T^{2} + 2555 T^{3} + 22872 T^{4} + 2555 p T^{5} + 239 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 152 T^{2} + 1018 T^{3} + 10798 T^{4} + 1018 p T^{5} + 152 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 91 T^{2} + 155 T^{3} + 2872 T^{4} + 155 p T^{5} + 91 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 24 T + 396 T^{2} - 4920 T^{3} + 49830 T^{4} - 4920 p T^{5} + 396 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 212 T^{2} + 1890 T^{3} + 25814 T^{4} + 1890 p T^{5} + 212 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 136 T^{2} + 1280 T^{3} + 16110 T^{4} + 1280 p T^{5} + 136 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 323 T^{2} + 2665 T^{3} + 37944 T^{4} + 2665 p T^{5} + 323 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.96533782031094690438592464026, −6.80242226107146098264162871806, −6.42483563759083864951338429728, −6.39995404412102598077196338387, −6.11416615276011873489365712737, −5.80017717445386214894354939702, −5.59807817860182320034012050669, −5.40075801317743351795784418323, −5.32454458171692868630858501985, −5.24990581075479334580043275570, −4.75895901353658390582609894907, −4.70565326152272854436276361643, −4.53558909667300863919179729368, −4.26364930555437060036039625304, −4.13594955239238313071468468496, −3.74180899562977542287645354573, −3.62498850997454474272934263897, −3.00859459784992687899187139590, −2.81570386255833249466026289327, −2.76802031154388122956489060334, −2.33642005094903179380584020249, −1.85430439698695992713526001106, −1.76686775996363392713797392868, −1.70812417877573680181152914734, −1.18682378364365594590547214346, 0, 0, 0, 0, 1.18682378364365594590547214346, 1.70812417877573680181152914734, 1.76686775996363392713797392868, 1.85430439698695992713526001106, 2.33642005094903179380584020249, 2.76802031154388122956489060334, 2.81570386255833249466026289327, 3.00859459784992687899187139590, 3.62498850997454474272934263897, 3.74180899562977542287645354573, 4.13594955239238313071468468496, 4.26364930555437060036039625304, 4.53558909667300863919179729368, 4.70565326152272854436276361643, 4.75895901353658390582609894907, 5.24990581075479334580043275570, 5.32454458171692868630858501985, 5.40075801317743351795784418323, 5.59807817860182320034012050669, 5.80017717445386214894354939702, 6.11416615276011873489365712737, 6.39995404412102598077196338387, 6.42483563759083864951338429728, 6.80242226107146098264162871806, 6.96533782031094690438592464026

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