Properties

Label 8-2010e4-1.1-c1e4-0-1
Degree $8$
Conductor $1.632\times 10^{13}$
Sign $1$
Analytic cond. $66357.9$
Root an. cond. $4.00623$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 4·3-s + 10·4-s − 4·5-s + 16·6-s + 7-s − 20·8-s + 10·9-s + 16·10-s − 3·11-s − 40·12-s + 2·13-s − 4·14-s + 16·15-s + 35·16-s − 2·17-s − 40·18-s + 2·19-s − 40·20-s − 4·21-s + 12·22-s − 12·23-s + 80·24-s + 10·25-s − 8·26-s − 20·27-s + 10·28-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 5·4-s − 1.78·5-s + 6.53·6-s + 0.377·7-s − 7.07·8-s + 10/3·9-s + 5.05·10-s − 0.904·11-s − 11.5·12-s + 0.554·13-s − 1.06·14-s + 4.13·15-s + 35/4·16-s − 0.485·17-s − 9.42·18-s + 0.458·19-s − 8.94·20-s − 0.872·21-s + 2.55·22-s − 2.50·23-s + 16.3·24-s + 2·25-s − 1.56·26-s − 3.84·27-s + 1.88·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 67^{4}\)
Sign: $1$
Analytic conductor: \(66357.9\)
Root analytic conductor: \(4.00623\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 67^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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$C_1$ \( ( 1 + T )^{4} \)
3$C_1$ \( ( 1 + T )^{4} \)
5$C_1$ \( ( 1 + T )^{4} \)
67$C_1$ \( ( 1 + T )^{4} \)
good7$C_2 \wr S_4$ \( 1 - T + 8 T^{2} - 9 T^{3} + 66 T^{4} - 9 p T^{5} + 8 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 3 T + 24 T^{2} + 67 T^{3} + 398 T^{4} + 67 p T^{5} + 24 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 2 T + 32 T^{2} - 50 T^{3} + 486 T^{4} - 50 p T^{5} + 32 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 2 T + 36 T^{2} + 22 T^{3} + 678 T^{4} + 22 p T^{5} + 36 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 2 T + 44 T^{2} - 34 T^{3} + 982 T^{4} - 34 p T^{5} + 44 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 12 T + 100 T^{2} + 624 T^{3} + 3094 T^{4} + 624 p T^{5} + 100 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 2 T + 64 T^{2} - 322 T^{3} + 1918 T^{4} - 322 p T^{5} + 64 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 10 T + 140 T^{2} - 878 T^{3} + 6646 T^{4} - 878 p T^{5} + 140 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 3 T + 122 T^{2} - 273 T^{3} + 6346 T^{4} - 273 p T^{5} + 122 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 92 T^{2} + 16 T^{3} + 4358 T^{4} + 16 p T^{5} + 92 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 6 T + 92 T^{2} + 518 T^{3} + 6006 T^{4} + 518 p T^{5} + 92 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 14 T + 116 T^{2} + 762 T^{3} + 4862 T^{4} + 762 p T^{5} + 116 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 10 T + 168 T^{2} + 1230 T^{3} + 12542 T^{4} + 1230 p T^{5} + 168 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 164 T^{2} + 16 T^{3} + 12566 T^{4} + 16 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 9 T + 254 T^{2} - 1599 T^{3} + 23518 T^{4} - 1599 p T^{5} + 254 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 9 T + 242 T^{2} + 1313 T^{3} + 22814 T^{4} + 1313 p T^{5} + 242 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 14 T + 248 T^{2} + 2274 T^{3} + 25134 T^{4} + 2274 p T^{5} + 248 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 12 T + 200 T^{2} - 992 T^{3} + 13686 T^{4} - 992 p T^{5} + 200 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 25 T + 324 T^{2} + 3793 T^{3} + 40262 T^{4} + 3793 p T^{5} + 324 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 9 T + 122 T^{2} + 783 T^{3} + 10410 T^{4} + 783 p T^{5} + 122 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 3 T + 368 T^{2} + 841 T^{3} + 52686 T^{4} + 841 p T^{5} + 368 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.04638203403860069670856862366, −6.60051035070603257863615518595, −6.53600082656102606678507907787, −6.44282755517497983071593833527, −6.42046436249885532765643778717, −6.00133534193021209741264634100, −5.71029150942541205664539627689, −5.57775943589656070171503565715, −5.52237848353146710178274141813, −4.85497104144803748572807269126, −4.80684426009838869038709551529, −4.74420173284237380040847525547, −4.61992845920438527234534832753, −3.88993766239842293547303410347, −3.88181068887422354669144568566, −3.67344504429635174646239847932, −3.63867274015453187377595458156, −2.79659753029482289526508617421, −2.74432168020512818391406893252, −2.49098331186921881387602977718, −2.42792012060599601742550466688, −1.46283408916974153530786996367, −1.40950598366389328212022683929, −1.26908834560176971307462856784, −1.22579544315174919631868493772, 0, 0, 0, 0, 1.22579544315174919631868493772, 1.26908834560176971307462856784, 1.40950598366389328212022683929, 1.46283408916974153530786996367, 2.42792012060599601742550466688, 2.49098331186921881387602977718, 2.74432168020512818391406893252, 2.79659753029482289526508617421, 3.63867274015453187377595458156, 3.67344504429635174646239847932, 3.88181068887422354669144568566, 3.88993766239842293547303410347, 4.61992845920438527234534832753, 4.74420173284237380040847525547, 4.80684426009838869038709551529, 4.85497104144803748572807269126, 5.52237848353146710178274141813, 5.57775943589656070171503565715, 5.71029150942541205664539627689, 6.00133534193021209741264634100, 6.42046436249885532765643778717, 6.44282755517497983071593833527, 6.53600082656102606678507907787, 6.60051035070603257863615518595, 7.04638203403860069670856862366

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