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Properties

Label 8-1815e4-1.1-c1e4-0-10
Degree $8$
Conductor $1.085\times 10^{13}$
Sign $1$
Analytic cond. $44117.9$
Root an. cond. $3.80694$
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  − 3·2-s + 4·3-s + 4-s + 4·5-s − 12·6-s − 6·7-s + 7·8-s + 10·9-s − 12·10-s + 4·12-s − 7·13-s + 18·14-s + 16·15-s − 10·16-s − 10·17-s − 30·18-s − 9·19-s + 4·20-s − 24·21-s − 3·23-s + 28·24-s + 10·25-s + 21·26-s + 20·27-s − 6·28-s − 15·29-s − 48·30-s + ⋯
L(s)  = 1  − 2.12·2-s + 2.30·3-s + 1/2·4-s + 1.78·5-s − 4.89·6-s − 2.26·7-s + 2.47·8-s + 10/3·9-s − 3.79·10-s + 1.15·12-s − 1.94·13-s + 4.81·14-s + 4.13·15-s − 5/2·16-s − 2.42·17-s − 7.07·18-s − 2.06·19-s + 0.894·20-s − 5.23·21-s − 0.625·23-s + 5.71·24-s + 2·25-s + 4.11·26-s + 3.84·27-s − 1.13·28-s − 2.78·29-s − 8.76·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(44117.9\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad3$C_1$ \( ( 1 - T )^{4} \)
5$C_1$ \( ( 1 - T )^{4} \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + p^{3} T^{2} + 7 p T^{3} + 23 T^{4} + 7 p^{2} T^{5} + p^{5} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 38 T^{2} + 129 T^{3} + 433 T^{4} + 129 p T^{5} + 38 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 62 T^{2} + 267 T^{3} + 1263 T^{4} + 267 p T^{5} + 62 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 84 T^{2} + 500 T^{3} + 2277 T^{4} + 500 p T^{5} + 84 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 98 T^{2} + 529 T^{3} + 3003 T^{4} + 529 p T^{5} + 98 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 37 T^{2} + 78 T^{3} + 1093 T^{4} + 78 p T^{5} + 37 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 120 T^{2} + 895 T^{3} + 5777 T^{4} + 895 p T^{5} + 120 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 13 T + 107 T^{2} + 602 T^{3} + 3463 T^{4} + 602 p T^{5} + 107 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 27 T^{2} + 120 T^{3} + 2801 T^{4} + 120 p T^{5} + 27 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 22 T + 302 T^{2} + 2913 T^{3} + 523 p T^{4} + 2913 p T^{5} + 302 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 62 T^{2} + 375 T^{3} + 1909 T^{4} + 375 p T^{5} + 62 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 96 T^{2} + 7 p T^{3} + 4637 T^{4} + 7 p^{2} T^{5} + 96 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 128 T^{2} - 775 T^{3} + 7309 T^{4} - 775 p T^{5} + 128 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 21 T + 353 T^{2} + 3746 T^{3} + 34103 T^{4} + 3746 p T^{5} + 353 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 156 T^{2} + 1147 T^{3} + 12741 T^{4} + 1147 p T^{5} + 156 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - T + 168 T^{2} - 89 T^{3} + 14153 T^{4} - 89 p T^{5} + 168 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 13 T + 227 T^{2} + 2382 T^{3} + 22433 T^{4} + 2382 p T^{5} + 227 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + T + 218 T^{2} + 135 T^{3} + 22101 T^{4} + 135 p T^{5} + 218 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 227 T^{2} - 662 T^{3} + 23215 T^{4} - 662 p T^{5} + 227 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 255 T^{2} + 600 T^{3} + 28593 T^{4} + 600 p T^{5} + 255 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 3 p T^{2} + 2360 T^{3} + 31893 T^{4} + 2360 p T^{5} + 3 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 22 T + 476 T^{2} + 6179 T^{3} + 72367 T^{4} + 6179 p T^{5} + 476 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.21159315312705769432783950087, −6.86678647848681114535183503858, −6.67929418610635142023527247627, −6.67054180861649211331605966271, −6.32019506983412606752399360870, −6.12795667028617477814404788934, −5.74521153341938863147988014215, −5.57164247031876126314180182629, −5.49330806667565481528107665989, −4.80968967207498363703640685356, −4.72058054585153469236559182509, −4.70987409377885338208287797661, −4.49791689313671462651541993067, −4.09733256502910899979989419632, −3.68457532705380612911412021582, −3.60135474851519164453162302319, −3.54985871074997107918499239656, −3.05234755630504328191036398206, −2.80513766790745663755595249178, −2.59561109431505567831266722434, −2.21095993453070364877829462839, −2.13266900401648292887654215753, −1.69392151046178218413921034882, −1.60241143058213505606575442354, −1.53653454582147124452331342164, 0, 0, 0, 0, 1.53653454582147124452331342164, 1.60241143058213505606575442354, 1.69392151046178218413921034882, 2.13266900401648292887654215753, 2.21095993453070364877829462839, 2.59561109431505567831266722434, 2.80513766790745663755595249178, 3.05234755630504328191036398206, 3.54985871074997107918499239656, 3.60135474851519164453162302319, 3.68457532705380612911412021582, 4.09733256502910899979989419632, 4.49791689313671462651541993067, 4.70987409377885338208287797661, 4.72058054585153469236559182509, 4.80968967207498363703640685356, 5.49330806667565481528107665989, 5.57164247031876126314180182629, 5.74521153341938863147988014215, 6.12795667028617477814404788934, 6.32019506983412606752399360870, 6.67054180861649211331605966271, 6.67929418610635142023527247627, 6.86678647848681114535183503858, 7.21159315312705769432783950087

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