Properties

Label 8-2352e4-1.1-c2e4-0-9
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $1.68690\times 10^{7}$
Root an. cond. $8.00545$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 24·5-s − 6·9-s + 48·13-s + 24·17-s + 264·25-s + 96·29-s − 72·41-s − 144·45-s − 88·53-s + 96·61-s + 1.15e3·65-s + 48·73-s + 27·81-s + 576·85-s − 120·89-s + 432·97-s − 72·101-s − 48·109-s + 56·113-s − 288·117-s + 76·121-s + 1.46e3·125-s + 127-s + 131-s + 137-s + 139-s + 2.30e3·145-s + ⋯
L(s)  = 1  + 24/5·5-s − 2/3·9-s + 3.69·13-s + 1.41·17-s + 10.5·25-s + 3.31·29-s − 1.75·41-s − 3.19·45-s − 1.66·53-s + 1.57·61-s + 17.7·65-s + 0.657·73-s + 1/3·81-s + 6.77·85-s − 1.34·89-s + 4.45·97-s − 0.712·101-s − 0.440·109-s + 0.495·113-s − 2.46·117-s + 0.628·121-s + 11.7·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 15.8·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.68690\times 10^{7}\)
Root analytic conductor: \(8.00545\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(49.07240279\)
\(L(\frac12)\) \(\approx\) \(49.07240279\)
\(L(2)\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( ( 1 - 12 T + 84 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 76 T^{2} - 10746 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} \)
13$D_{4}$ \( ( 1 - 24 T + 384 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 - 12 T + 564 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 506 T^{2} + p^{4} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 20 T^{2} + 186534 T^{4} + 20 p^{4} T^{6} + p^{8} T^{8} \)
29$D_{4}$ \( ( 1 - 48 T + 2186 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 140 T^{2} + 1478694 T^{4} + 140 p^{4} T^{6} + p^{8} T^{8} \)
37$C_2^2$ \( ( 1 + 1586 T^{2} + p^{4} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 36 T + 3588 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 1732 T^{2} + 3440358 T^{4} - 1732 p^{4} T^{6} + p^{8} T^{8} \)
47$D_4\times C_2$ \( 1 - 8020 T^{2} + 25673574 T^{4} - 8020 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 + 44 T + 3510 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 5332 T^{2} + 13053126 T^{4} - 5332 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 - 48 T + 4656 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 16900 T^{2} + 111538854 T^{4} - 16900 p^{4} T^{6} + p^{8} T^{8} \)
71$D_4\times C_2$ \( 1 - 11308 T^{2} + 69354150 T^{4} - 11308 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 24 T + 8352 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 16708 T^{2} + 131100678 T^{4} - 16708 p^{4} T^{6} + p^{8} T^{8} \)
83$D_4\times C_2$ \( 1 - 23428 T^{2} + 227987238 T^{4} - 23428 p^{4} T^{6} + p^{8} T^{8} \)
89$D_{4}$ \( ( 1 + 60 T + 180 p T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 216 T + 30240 T^{2} - 216 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.20281113588544159851415962540, −5.78635057915448746967028664781, −5.76480388482676885263663568569, −5.72243473893383734962647483720, −5.70799624459140453850516729126, −5.20782543475960376107016141642, −5.11439917067438083198265014375, −4.86893788847150908126870829394, −4.70981314299788384203623319762, −4.23848230285363205337688645070, −4.16592670963345229995888599773, −3.63820936730699563709956282936, −3.35949964832535468312225887976, −3.31501368494993253053685164023, −3.20175357904545609349346048460, −2.78486865042626087058296151810, −2.48007722872428867990885393531, −2.30086632155656766348098170871, −2.01837130752060678779952259649, −1.77931613242939719573208379435, −1.48243426222085940900080785820, −1.40925421513066157818990240557, −1.12649671029041723147494494105, −0.805721318105038327472245186024, −0.55684014476926840054658952463, 0.55684014476926840054658952463, 0.805721318105038327472245186024, 1.12649671029041723147494494105, 1.40925421513066157818990240557, 1.48243426222085940900080785820, 1.77931613242939719573208379435, 2.01837130752060678779952259649, 2.30086632155656766348098170871, 2.48007722872428867990885393531, 2.78486865042626087058296151810, 3.20175357904545609349346048460, 3.31501368494993253053685164023, 3.35949964832535468312225887976, 3.63820936730699563709956282936, 4.16592670963345229995888599773, 4.23848230285363205337688645070, 4.70981314299788384203623319762, 4.86893788847150908126870829394, 5.11439917067438083198265014375, 5.20782543475960376107016141642, 5.70799624459140453850516729126, 5.72243473893383734962647483720, 5.76480388482676885263663568569, 5.78635057915448746967028664781, 6.20281113588544159851415962540

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