Properties

Label 8-1805e4-1.1-c1e4-0-3
Degree $8$
Conductor $1.061\times 10^{13}$
Sign $1$
Analytic cond. $43153.6$
Root an. cond. $3.79644$
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  − 3·4-s − 4·5-s − 8·7-s − 7·9-s − 10·11-s + 12·17-s + 12·20-s − 6·23-s + 10·25-s + 24·28-s + 32·35-s + 21·36-s + 6·43-s + 30·44-s + 28·45-s + 12·47-s + 22·49-s + 40·55-s − 20·61-s + 56·63-s + 15·64-s − 36·68-s − 4·73-s + 80·77-s + 20·81-s − 44·83-s − 48·85-s + ⋯
L(s)  = 1  − 3/2·4-s − 1.78·5-s − 3.02·7-s − 7/3·9-s − 3.01·11-s + 2.91·17-s + 2.68·20-s − 1.25·23-s + 2·25-s + 4.53·28-s + 5.40·35-s + 7/2·36-s + 0.914·43-s + 4.52·44-s + 4.17·45-s + 1.75·47-s + 22/7·49-s + 5.39·55-s − 2.56·61-s + 7.05·63-s + 15/8·64-s − 4.36·68-s − 0.468·73-s + 9.11·77-s + 20/9·81-s − 4.82·83-s − 5.20·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(43153.6\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1805} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 5^{4} \cdot 19^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad5$C_1$ \( ( 1 + T )^{4} \)
19 \( 1 \)
good2$C_2^2:C_4$ \( 1 + 3 T^{2} + 9 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \)
3$C_2^2:C_4$ \( 1 + 7 T^{2} + 29 T^{4} + 7 p^{2} T^{6} + p^{4} T^{8} \)
7$D_{4}$ \( ( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
11$D_{4}$ \( ( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2:C_4$ \( 1 + 42 T^{2} + 759 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_{4}$ \( ( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2:C_4$ \( 1 + 31 T^{2} + 721 T^{4} + 31 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2^2:C_4$ \( 1 + 119 T^{2} + 5461 T^{4} + 119 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2^2:C_4$ \( 1 + 63 T^{2} + 2529 T^{4} + 63 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2:C_4$ \( 1 + 114 T^{2} + 6111 T^{4} + 114 p^{2} T^{6} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 - 3 T + 87 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_{4}$ \( ( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
53$C_2^2:C_4$ \( 1 + 87 T^{2} + 3729 T^{4} + 87 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2:C_4$ \( 1 - 89 T^{2} + 7741 T^{4} - 89 p^{2} T^{6} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 + 10 T + 67 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2:C_4$ \( 1 + 98 T^{2} + 8959 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2:C_4$ \( 1 + 114 T^{2} + 6111 T^{4} + 114 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
79$C_2^2:C_4$ \( 1 + 216 T^{2} + 23646 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 + 22 T + 282 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \)
89$C_2^2:C_4$ \( 1 + 226 T^{2} + 28591 T^{4} + 226 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2:C_4$ \( 1 + 363 T^{2} + 51729 T^{4} + 363 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.30512940832295648735984909465, −6.72097173069887569669087051926, −6.51987916468680208258760091391, −6.28335492649416740627705381164, −6.18248332717412165890682506548, −5.72283598798135121858743076746, −5.71174689487335185254999396649, −5.61826140609851748325734354868, −5.49463689292032270638611012922, −5.11876385316569173526160475700, −4.72457422432589729077195513728, −4.68446871552398458475813387004, −4.58878216447706277760324531813, −4.05101507018373445083029355885, −3.73734678939632829291162611423, −3.65461680116967875928998312304, −3.56377211486846548284762013262, −3.22381970096534953046652697956, −2.94670681704639266892164841900, −2.69808227559460142253821564047, −2.67824812665464001826250491565, −2.63935566456426318279151265882, −1.98623982132437163945892345415, −1.14823418666610834223641204809, −0.877302181015217455908674217002, 0, 0, 0, 0, 0.877302181015217455908674217002, 1.14823418666610834223641204809, 1.98623982132437163945892345415, 2.63935566456426318279151265882, 2.67824812665464001826250491565, 2.69808227559460142253821564047, 2.94670681704639266892164841900, 3.22381970096534953046652697956, 3.56377211486846548284762013262, 3.65461680116967875928998312304, 3.73734678939632829291162611423, 4.05101507018373445083029355885, 4.58878216447706277760324531813, 4.68446871552398458475813387004, 4.72457422432589729077195513728, 5.11876385316569173526160475700, 5.49463689292032270638611012922, 5.61826140609851748325734354868, 5.71174689487335185254999396649, 5.72283598798135121858743076746, 6.18248332717412165890682506548, 6.28335492649416740627705381164, 6.51987916468680208258760091391, 6.72097173069887569669087051926, 7.30512940832295648735984909465

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