Properties

Label 8-6864e4-1.1-c1e4-0-1
Degree $8$
Conductor $2.220\times 10^{15}$
Sign $1$
Analytic cond. $9.02438\times 10^{6}$
Root an. cond. $7.40333$
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  + 4·3-s + 5-s + 7-s + 10·9-s − 4·11-s − 4·13-s + 4·15-s − 6·17-s + 4·21-s + 9·23-s − 9·25-s + 20·27-s − 29-s + 8·31-s − 16·33-s + 35-s − 2·37-s − 16·39-s + 9·41-s + 5·43-s + 10·45-s + 22·47-s − 9·49-s − 24·51-s + 8·53-s − 4·55-s − 59-s + ⋯
L(s)  = 1  + 2.30·3-s + 0.447·5-s + 0.377·7-s + 10/3·9-s − 1.20·11-s − 1.10·13-s + 1.03·15-s − 1.45·17-s + 0.872·21-s + 1.87·23-s − 9/5·25-s + 3.84·27-s − 0.185·29-s + 1.43·31-s − 2.78·33-s + 0.169·35-s − 0.328·37-s − 2.56·39-s + 1.40·41-s + 0.762·43-s + 1.49·45-s + 3.20·47-s − 9/7·49-s − 3.36·51-s + 1.09·53-s − 0.539·55-s − 0.130·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(9.02438\times 10^{6}\)
Root analytic conductor: \(7.40333\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6864} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(21.20817063\)
\(L(\frac12)\) \(\approx\) \(21.20817063\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{4} \)
11$C_1$ \( ( 1 + T )^{4} \)
13$C_1$ \( ( 1 + T )^{4} \)
good5$C_2 \wr S_4$ \( 1 - T + 2 p T^{2} - p T^{3} + 48 T^{4} - p^{2} T^{5} + 2 p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 - T + 10 T^{2} - 13 T^{3} + 10 p T^{4} - 13 p T^{5} + 10 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 6 T + 46 T^{2} + 216 T^{3} + 1218 T^{4} + 216 p T^{5} + 46 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 28 T^{2} - 108 T^{3} + 414 T^{4} - 108 p T^{5} + 28 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 9 T + 76 T^{2} - 453 T^{3} + 2730 T^{4} - 453 p T^{5} + 76 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + T + 62 T^{2} + 259 T^{3} + 1780 T^{4} + 259 p T^{5} + 62 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 8 T + 120 T^{2} - 718 T^{3} + 5510 T^{4} - 718 p T^{5} + 120 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 2 T + 108 T^{2} + 142 T^{3} + 5222 T^{4} + 142 p T^{5} + 108 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 9 T + 176 T^{2} - 1071 T^{3} + 11034 T^{4} - 1071 p T^{5} + 176 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 5 T + 54 T^{2} - 7 p T^{3} + 3620 T^{4} - 7 p^{2} T^{5} + 54 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 22 T + 296 T^{2} - 2950 T^{3} + 22510 T^{4} - 2950 p T^{5} + 296 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 8 T + 124 T^{2} - 8 p T^{3} + 5910 T^{4} - 8 p^{2} T^{5} + 124 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + T + 64 T^{2} + 305 T^{3} + 6078 T^{4} + 305 p T^{5} + 64 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 11 T + 108 T^{2} + 781 T^{3} + 4550 T^{4} + 781 p T^{5} + 108 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 13 T + 180 T^{2} - 1235 T^{3} + 11516 T^{4} - 1235 p T^{5} + 180 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 8 T + 196 T^{2} - 856 T^{3} + 16134 T^{4} - 856 p T^{5} + 196 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 3 T + 64 T^{2} + 369 T^{3} + 2754 T^{4} + 369 p T^{5} + 64 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 6 T + 174 T^{2} - 1320 T^{3} + 15398 T^{4} - 1320 p T^{5} + 174 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 18 T + 364 T^{2} - 4002 T^{3} + 45078 T^{4} - 4002 p T^{5} + 364 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 8 T + 352 T^{2} - 2002 T^{3} + 46590 T^{4} - 2002 p T^{5} + 352 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 10 T + 148 T^{2} - 334 T^{3} + 5830 T^{4} - 334 p T^{5} + 148 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.74497649761612070621231836771, −5.27289291138736477083168630756, −5.08275836219410314073456125978, −5.05565054609485829651572473322, −4.72023130585570917220570456472, −4.64820317927332447274639330747, −4.36320732904890544979701464020, −4.36176015423648488926371019166, −4.10335085241219726420427450509, −3.70104113418123551034458910891, −3.61572676144297510528591599137, −3.60375309521087566139096355299, −3.15262503568123356316598015924, −2.88524984795657189862372587537, −2.81200120341979243166919960193, −2.62385752105451812353665252685, −2.44074017873941600848111925102, −2.13678824245385146321483853367, −2.01779302182352517373450885736, −2.00844397431474141938390203981, −1.73115356986566539217591542723, −1.14432576572104660776279558679, −0.923436639841302740364866937713, −0.62022020260467284746473901043, −0.44786720538132208449908148316, 0.44786720538132208449908148316, 0.62022020260467284746473901043, 0.923436639841302740364866937713, 1.14432576572104660776279558679, 1.73115356986566539217591542723, 2.00844397431474141938390203981, 2.01779302182352517373450885736, 2.13678824245385146321483853367, 2.44074017873941600848111925102, 2.62385752105451812353665252685, 2.81200120341979243166919960193, 2.88524984795657189862372587537, 3.15262503568123356316598015924, 3.60375309521087566139096355299, 3.61572676144297510528591599137, 3.70104113418123551034458910891, 4.10335085241219726420427450509, 4.36176015423648488926371019166, 4.36320732904890544979701464020, 4.64820317927332447274639330747, 4.72023130585570917220570456472, 5.05565054609485829651572473322, 5.08275836219410314073456125978, 5.27289291138736477083168630756, 5.74497649761612070621231836771

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