Properties

Label 8-13e8-1.1-c3e4-0-8
Degree $8$
Conductor $815730721$
Sign $1$
Analytic cond. $9885.78$
Root an. cond. $3.15774$
Motivic weight $3$
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  + 14·3-s + 4·4-s + 103·9-s + 56·12-s + 64·16-s + 54·17-s + 114·23-s − 116·25-s + 826·27-s + 138·29-s + 412·36-s − 170·43-s + 896·48-s + 179·49-s + 756·51-s + 1.70e3·53-s + 34·61-s + 704·64-s + 216·68-s + 1.59e3·69-s − 1.62e3·75-s − 4.97e3·79-s + 7.00e3·81-s + 1.93e3·87-s + 456·92-s − 464·100-s + 3.91e3·101-s + ⋯
L(s)  = 1  + 2.69·3-s + 1/2·4-s + 3.81·9-s + 1.34·12-s + 16-s + 0.770·17-s + 1.03·23-s − 0.927·25-s + 5.88·27-s + 0.883·29-s + 1.90·36-s − 0.602·43-s + 2.69·48-s + 0.521·49-s + 2.07·51-s + 4.41·53-s + 0.0713·61-s + 11/8·64-s + 0.385·68-s + 2.78·69-s − 2.50·75-s − 7.08·79-s + 9.60·81-s + 2.38·87-s + 0.516·92-s − 0.463·100-s + 3.85·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(13^{8}\)
Sign: $1$
Analytic conductor: \(9885.78\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 13^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(16.80325619\)
\(L(\frac12)\) \(\approx\) \(16.80325619\)
\(L(\frac{5}{2})\) 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)$
bad13 \( 1 \)
good2$C_2^3$ \( 1 - p^{2} T^{2} - 3 p^{4} T^{4} - p^{8} T^{6} + p^{12} T^{8} \)
3$C_2^2$ \( ( 1 - 7 T + 22 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 + 58 T^{2} + p^{6} T^{4} )^{2} \)
7$C_2^3$ \( 1 - 179 T^{2} - 85608 T^{4} - 179 p^{6} T^{6} + p^{12} T^{8} \)
11$C_2^3$ \( 1 - 2155 T^{2} + 2872464 T^{4} - 2155 p^{6} T^{6} + p^{12} T^{8} \)
17$C_2^2$ \( ( 1 - 27 T - 4184 T^{2} - 27 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
19$C_2^3$ \( 1 - 5915 T^{2} - 12058656 T^{4} - 5915 p^{6} T^{6} + p^{12} T^{8} \)
23$C_2^2$ \( ( 1 - 57 T - 8918 T^{2} - 57 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 69 T - 19628 T^{2} - 69 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 54290 T^{2} + p^{6} T^{4} )^{2} \)
37$C_2^3$ \( 1 - 99719 T^{2} + 7378152552 T^{4} - 99719 p^{6} T^{6} + p^{12} T^{8} \)
41$C_2^3$ \( 1 + 16745 T^{2} - 4469709216 T^{4} + 16745 p^{6} T^{6} + p^{12} T^{8} \)
43$C_2^2$ \( ( 1 + 85 T - 72282 T^{2} + 85 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 90034 T^{2} + p^{6} T^{4} )^{2} \)
53$C_2$ \( ( 1 - 426 T + p^{3} T^{2} )^{4} \)
59$C_2^3$ \( 1 - 410395 T^{2} + 126243522384 T^{4} - 410395 p^{6} T^{6} + p^{12} T^{8} \)
61$C_2^2$ \( ( 1 - 17 T - 226692 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
67$C_2^3$ \( 1 - 574451 T^{2} + 239535569232 T^{4} - 574451 p^{6} T^{6} + p^{12} T^{8} \)
71$C_2^3$ \( 1 - 375115 T^{2} + 12610979304 T^{4} - 375115 p^{6} T^{6} + p^{12} T^{8} \)
73$C_2^2$ \( ( 1 - 231166 T^{2} + p^{6} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 1244 T + p^{3} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 962026 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2^3$ \( 1 - 1315951 T^{2} + 1234745743440 T^{4} - 1315951 p^{6} T^{6} + p^{12} T^{8} \)
97$C_2^3$ \( 1 - 300239 T^{2} - 742828547808 T^{4} - 300239 p^{6} T^{6} + p^{12} T^{8} \)
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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

−8.783186784655053790692296812578, −8.546305185147663937398107420501, −8.418041462657932276202687146163, −8.367453648465941067597285637160, −7.81685575874874361855510550476, −7.47224532272548667089043245728, −7.34105417824589730900377208084, −7.10221872891256214390298377305, −6.89820921126313904293861866205, −6.38352627037604110932431769694, −6.21017703840753424181191599403, −5.57835254209185426275708712413, −5.35242030642084998889721545685, −5.21215179297313945104997192734, −4.50593432421718688574708695636, −4.10360630340555716848383515633, −4.03323144945382492288956548147, −3.55471853931236530034756333824, −3.12004335684016734339341771507, −2.75587944190217775525305487949, −2.69994388055419531329030054709, −2.36986999567970864789516699977, −1.53757732922171221992105764468, −1.31248309036038851134004631207, −0.71893077149200097815950516107, 0.71893077149200097815950516107, 1.31248309036038851134004631207, 1.53757732922171221992105764468, 2.36986999567970864789516699977, 2.69994388055419531329030054709, 2.75587944190217775525305487949, 3.12004335684016734339341771507, 3.55471853931236530034756333824, 4.03323144945382492288956548147, 4.10360630340555716848383515633, 4.50593432421718688574708695636, 5.21215179297313945104997192734, 5.35242030642084998889721545685, 5.57835254209185426275708712413, 6.21017703840753424181191599403, 6.38352627037604110932431769694, 6.89820921126313904293861866205, 7.10221872891256214390298377305, 7.34105417824589730900377208084, 7.47224532272548667089043245728, 7.81685575874874361855510550476, 8.367453648465941067597285637160, 8.418041462657932276202687146163, 8.546305185147663937398107420501, 8.783186784655053790692296812578

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