Properties

Label 8-3200e4-1.1-c1e4-0-23
Degree $8$
Conductor $1.049\times 10^{14}$
Sign $1$
Analytic cond. $426293.$
Root an. cond. $5.05491$
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  − 16·17-s + 32·41-s − 16·49-s + 16·73-s − 18·81-s − 8·89-s + 16·97-s + 64·113-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 3.88·17-s + 4.99·41-s − 2.28·49-s + 1.87·73-s − 2·81-s − 0.847·89-s + 1.62·97-s + 6.02·113-s − 0.363·121-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 + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(426293.\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.535056141\)
\(L(\frac12)\) \(\approx\) \(3.535056141\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
97$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
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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

−6.08797486155196423425015904944, −5.99068701587077288274577183458, −5.87634819247458266645791226122, −5.67327208512409853972541105525, −5.30771855816824130925688569172, −4.93037169443121535536541115174, −4.78763567242103038988951752184, −4.70657617482620261566458637429, −4.64129173711993029500116854503, −4.23666028465203257760912094100, −4.07341132925047361482517083902, −3.94470978003321719287294247891, −3.87649226425293523423832340623, −3.25919946586406042264397123034, −3.18162717407602784298173151095, −2.87261124763231417361232692252, −2.63590278244596198475322251602, −2.49419772232160104124188838813, −2.06079137230374219401437132078, −2.05052485174579993904240380620, −1.73982622126437590290783319049, −1.50429610665218182842673689060, −0.74456486605857232212256740680, −0.64787843041101306061059246520, −0.39458595484062388352807966595, 0.39458595484062388352807966595, 0.64787843041101306061059246520, 0.74456486605857232212256740680, 1.50429610665218182842673689060, 1.73982622126437590290783319049, 2.05052485174579993904240380620, 2.06079137230374219401437132078, 2.49419772232160104124188838813, 2.63590278244596198475322251602, 2.87261124763231417361232692252, 3.18162717407602784298173151095, 3.25919946586406042264397123034, 3.87649226425293523423832340623, 3.94470978003321719287294247891, 4.07341132925047361482517083902, 4.23666028465203257760912094100, 4.64129173711993029500116854503, 4.70657617482620261566458637429, 4.78763567242103038988951752184, 4.93037169443121535536541115174, 5.30771855816824130925688569172, 5.67327208512409853972541105525, 5.87634819247458266645791226122, 5.99068701587077288274577183458, 6.08797486155196423425015904944

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