Properties

Label 8-1216e4-1.1-c1e4-0-5
Degree $8$
Conductor $2.186\times 10^{12}$
Sign $1$
Analytic cond. $8888.79$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 5-s + 7-s − 9-s + 7·11-s − 10·13-s − 2·15-s + 5·17-s + 4·19-s + 2·21-s + 8·23-s − 25-s − 2·27-s − 10·29-s + 6·31-s + 14·33-s − 35-s − 14·37-s − 20·39-s − 2·41-s + 3·43-s + 45-s + 3·47-s − 10·49-s + 10·51-s − 4·53-s − 7·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 0.377·7-s − 1/3·9-s + 2.11·11-s − 2.77·13-s − 0.516·15-s + 1.21·17-s + 0.917·19-s + 0.436·21-s + 1.66·23-s − 1/5·25-s − 0.384·27-s − 1.85·29-s + 1.07·31-s + 2.43·33-s − 0.169·35-s − 2.30·37-s − 3.20·39-s − 0.312·41-s + 0.457·43-s + 0.149·45-s + 0.437·47-s − 1.42·49-s + 1.40·51-s − 0.549·53-s − 0.943·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(8888.79\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.006231920\)
\(L(\frac12)\) \(\approx\) \(3.006231920\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 - T )^{4} \)
good3$D_{4}$ \( ( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + T + 2 T^{2} - 9 T^{3} + 2 T^{4} - 9 p T^{5} + 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - T + 11 T^{2} - 40 T^{3} + 60 T^{4} - 40 p T^{5} + 11 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 4 p T^{2} - 203 T^{3} + 742 T^{4} - 203 p T^{5} + 4 p^{3} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 59 T^{2} + 232 T^{3} + 852 T^{4} + 232 p T^{5} + 59 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + p T^{2} - 114 T^{3} + 698 T^{4} - 114 p T^{5} + p^{3} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 75 T^{2} - 420 T^{3} + 2456 T^{4} - 420 p T^{5} + 75 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 123 T^{2} + 712 T^{3} + 5108 T^{4} + 712 p T^{5} + 123 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 68 T^{2} - 142 T^{3} + 1782 T^{4} - 142 p T^{5} + 68 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 152 T^{2} + 1018 T^{3} + 7006 T^{4} + 1018 p T^{5} + 152 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 92 T^{2} + 54 T^{3} + 4694 T^{4} + 54 p T^{5} + 92 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 140 T^{2} - 291 T^{3} + 8278 T^{4} - 291 p T^{5} + 140 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 80 T^{2} - 247 T^{3} + 3038 T^{4} - 247 p T^{5} + 80 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 143 T^{2} + 252 T^{3} + 9032 T^{4} + 252 p T^{5} + 143 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 321 T^{2} - 3390 T^{3} + 30764 T^{4} - 3390 p T^{5} + 321 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 178 T^{2} - 521 T^{3} + 14618 T^{4} - 521 p T^{5} + 178 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 193 T^{2} - 1006 T^{3} + 16060 T^{4} - 1006 p T^{5} + 193 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 30 T + 552 T^{2} - 6814 T^{3} + 65870 T^{4} - 6814 p T^{5} + 552 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 245 T^{2} - 1662 T^{3} + 25114 T^{4} - 1662 p T^{5} + 245 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 216 T^{2} + 1082 T^{3} + 18510 T^{4} + 1082 p T^{5} + 216 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 216 T^{2} - 212 T^{3} + 21054 T^{4} - 212 p T^{5} + 216 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 152 T^{2} + 224 T^{3} - 4082 T^{4} + 224 p T^{5} + 152 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 328 T^{2} + 1658 T^{3} + 45422 T^{4} + 1658 p T^{5} + 328 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.99034840590051602903020580813, −6.70881497113278522300333750110, −6.68016368521213508621002392349, −6.56282440371279163166581079474, −6.17197129387899924194818303283, −5.60729607740898027930719918903, −5.38934180538649206058523583571, −5.37799706097445190396062840311, −5.33765120567546592002073051100, −4.92573797800613736282758985059, −4.73211850114360463058878088810, −4.41445634409341867414980068449, −4.16394025055850417656785202711, −3.75250334344436521117111165906, −3.59107610457235071588320029686, −3.47557769501295184170228693616, −3.30278924354539835556687382360, −2.77990280888157109607639758188, −2.66390732337425207386376753926, −2.34536741288858229298641641735, −2.13620304908082963292739725866, −1.68092204479684283311066936891, −1.19533803010128349088953861527, −1.08078416561479399268482644582, −0.33524606563212228059083201473, 0.33524606563212228059083201473, 1.08078416561479399268482644582, 1.19533803010128349088953861527, 1.68092204479684283311066936891, 2.13620304908082963292739725866, 2.34536741288858229298641641735, 2.66390732337425207386376753926, 2.77990280888157109607639758188, 3.30278924354539835556687382360, 3.47557769501295184170228693616, 3.59107610457235071588320029686, 3.75250334344436521117111165906, 4.16394025055850417656785202711, 4.41445634409341867414980068449, 4.73211850114360463058878088810, 4.92573797800613736282758985059, 5.33765120567546592002073051100, 5.37799706097445190396062840311, 5.38934180538649206058523583571, 5.60729607740898027930719918903, 6.17197129387899924194818303283, 6.56282440371279163166581079474, 6.68016368521213508621002392349, 6.70881497113278522300333750110, 6.99034840590051602903020580813

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