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Properties

Label 8-10e16-1.1-c1e4-0-2
Degree $8$
Conductor $1.000\times 10^{16}$
Sign $1$
Analytic cond. $4.06544\times 10^{7}$
Root an. cond. $8.93590$
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  + 3-s + 2·7-s − 2·9-s + 2·11-s − 11·13-s − 12·17-s + 5·19-s + 2·21-s − 4·23-s + 15·29-s + 12·31-s + 2·33-s − 12·37-s − 11·39-s + 13·41-s + 6·43-s + 2·47-s − 13·49-s − 12·51-s − 11·53-s + 5·57-s + 8·61-s − 4·63-s + 22·67-s − 4·69-s + 22·71-s − 21·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s − 2/3·9-s + 0.603·11-s − 3.05·13-s − 2.91·17-s + 1.14·19-s + 0.436·21-s − 0.834·23-s + 2.78·29-s + 2.15·31-s + 0.348·33-s − 1.97·37-s − 1.76·39-s + 2.03·41-s + 0.914·43-s + 0.291·47-s − 1.85·49-s − 1.68·51-s − 1.51·53-s + 0.662·57-s + 1.02·61-s − 0.503·63-s + 2.68·67-s − 0.481·69-s + 2.61·71-s − 2.45·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(4.06544\times 10^{7}\)
Root analytic conductor: \(8.93590\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{16} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.185839268\)
\(L(\frac12)\) \(\approx\) \(6.185839268\)
\(L(\frac{3}{2})\) 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$C_2^2:C_4$ \( 1 - T + p T^{2} - 5 T^{3} + 16 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 17 T^{2} - 30 T^{3} + 156 T^{4} - 30 p T^{5} + 17 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 3 p T^{2} - 54 T^{3} + 500 T^{4} - 54 p T^{5} + 3 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 6 p T^{2} + 370 T^{3} + 1491 T^{4} + 370 p T^{5} + 6 p^{3} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 107 T^{2} + 610 T^{3} + 2951 T^{4} + 610 p T^{5} + 107 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2:C_4$ \( 1 - 5 T + 41 T^{2} - 265 T^{3} + 916 T^{4} - 265 p T^{5} + 41 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 63 T^{2} + 280 T^{3} + 1856 T^{4} + 280 p T^{5} + 63 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 15 T + 186 T^{2} - 1410 T^{3} + 9111 T^{4} - 1410 p T^{5} + 186 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 123 T^{2} - 684 T^{3} + 4440 T^{4} - 684 p T^{5} + 123 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 167 T^{2} + 1230 T^{3} + 9691 T^{4} + 1230 p T^{5} + 167 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 13 T + 198 T^{2} - 1576 T^{3} + 12785 T^{4} - 1576 p T^{5} + 198 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 133 T^{2} - 710 T^{3} + 7916 T^{4} - 710 p T^{5} + 133 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 177 T^{2} - 270 T^{3} + 12236 T^{4} - 270 p T^{5} + 177 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 163 T^{2} + 1285 T^{3} + 11916 T^{4} + 1285 p T^{5} + 163 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 96 T^{2} + 560 T^{3} + 4046 T^{4} + 560 p T^{5} + 96 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 143 T^{2} - 506 T^{3} + 8295 T^{4} - 506 p T^{5} + 143 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 22 T + 337 T^{2} - 3850 T^{3} + 35236 T^{4} - 3850 p T^{5} + 337 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 22 T + 413 T^{2} - 4854 T^{3} + 48500 T^{4} - 4854 p T^{5} + 413 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 21 T + 373 T^{2} + 4385 T^{3} + 42716 T^{4} + 4385 p T^{5} + 373 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 296 T^{2} - 2130 T^{3} + 33966 T^{4} - 2130 p T^{5} + 296 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 463 T^{2} + 5520 T^{3} + 59416 T^{4} + 5520 p T^{5} + 463 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 191 T^{2} - 165 T^{3} + 15056 T^{4} - 165 p T^{5} + 191 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 + 11 T + 193 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−5.37135069101951160079606405748, −5.16242252188063419691036240488, −4.90103952945369901317431360356, −4.71983446957965921618172746142, −4.52172753209789870456944484047, −4.48830258691185234389943108581, −4.39917272164869624707321581506, −4.19476388804737672948515038172, −4.16162317719659061399756602165, −3.64391201609919940650966565226, −3.36503887760399601642851768596, −3.21625065019350257804145380109, −3.10962285002614148066206569058, −2.75865822108879107173767190997, −2.61910747585968616596917198427, −2.56520804021324090931854373750, −2.35185639626706866947760723691, −2.19303675628893038256483412883, −1.89841951596225293247151635924, −1.62567419204546333727487084347, −1.51767080077651283610018525890, −1.06089750767912378933815704961, −0.65918236881154728129738720068, −0.46684339968310399338868529151, −0.38301194296712549894306772612, 0.38301194296712549894306772612, 0.46684339968310399338868529151, 0.65918236881154728129738720068, 1.06089750767912378933815704961, 1.51767080077651283610018525890, 1.62567419204546333727487084347, 1.89841951596225293247151635924, 2.19303675628893038256483412883, 2.35185639626706866947760723691, 2.56520804021324090931854373750, 2.61910747585968616596917198427, 2.75865822108879107173767190997, 3.10962285002614148066206569058, 3.21625065019350257804145380109, 3.36503887760399601642851768596, 3.64391201609919940650966565226, 4.16162317719659061399756602165, 4.19476388804737672948515038172, 4.39917272164869624707321581506, 4.48830258691185234389943108581, 4.52172753209789870456944484047, 4.71983446957965921618172746142, 4.90103952945369901317431360356, 5.16242252188063419691036240488, 5.37135069101951160079606405748

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