↑

Properties

Label 8-45e4-1.1-c4e4-0-1
Degree $8$
Conductor $4100625$
Sign $1$
Analytic cond. $468.195$
Root an. cond. $2.15676$
Motivic weight $4$
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  + 18·4-s − 48·7-s + 808·13-s + 91·16-s − 1.76e3·19-s − 250·25-s − 864·28-s − 1.65e3·31-s + 344·37-s + 5.77e3·43-s + 656·49-s + 1.45e4·52-s − 2.01e4·61-s + 2.05e3·64-s − 1.58e3·67-s + 88·73-s − 3.16e4·76-s + 1.77e4·79-s − 3.87e4·91-s + 2.74e4·97-s − 4.50e3·100-s − 400·103-s − 6.52e3·109-s − 4.36e3·112-s + 4.74e4·121-s − 2.98e4·124-s + 127-s + ⋯
L(s)  = 1  + 9/8·4-s − 0.979·7-s + 4.78·13-s + 0.355·16-s − 4.87·19-s − 2/5·25-s − 1.10·28-s − 1.72·31-s + 0.251·37-s + 3.12·43-s + 0.273·49-s + 5.37·52-s − 5.42·61-s + 0.500·64-s − 0.352·67-s + 0.0165·73-s − 5.48·76-s + 2.84·79-s − 4.68·91-s + 2.91·97-s − 0.449·100-s − 0.0377·103-s − 0.548·109-s − 0.348·112-s + 3.24·121-s − 1.93·124-s + 6.20e−5·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(4100625\)    =    \(3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(468.195\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 4100625,\ (\ :2, 2, 2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.881954431\)
\(L(\frac12)\) \(\approx\) \(2.881954431\)
\(L(3)\) 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)$
bad3 \( 1 \)
5$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
good2$D_4\times C_2$ \( 1 - 9 p T^{2} + 233 T^{4} - 9 p^{9} T^{6} + p^{16} T^{8} \)
7$D_{4}$ \( ( 1 + 24 T + 536 T^{2} + 24 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 47448 T^{2} + 962099378 T^{4} - 47448 p^{8} T^{6} + p^{16} T^{8} \)
13$D_{4}$ \( ( 1 - 404 T + 95676 T^{2} - 404 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 138948 T^{2} + 10035485318 T^{4} - 138948 p^{8} T^{6} + p^{16} T^{8} \)
19$D_{4}$ \( ( 1 + 880 T + 453882 T^{2} + 880 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 254860 T^{2} + 172735907622 T^{4} - 254860 p^{8} T^{6} + p^{16} T^{8} \)
29$D_4\times C_2$ \( 1 - 1417500 T^{2} + 1274204988422 T^{4} - 1417500 p^{8} T^{6} + p^{16} T^{8} \)
31$D_{4}$ \( ( 1 + 828 T + 848798 T^{2} + 828 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 172 T + 2066508 T^{2} - 172 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 4942080 T^{2} + 12337515177602 T^{4} - 4942080 p^{8} T^{6} + p^{16} T^{8} \)
43$D_{4}$ \( ( 1 - 2888 T + 8193738 T^{2} - 2888 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 12953940 T^{2} + 88587954654182 T^{4} - 12953940 p^{8} T^{6} + p^{16} T^{8} \)
53$D_4\times C_2$ \( 1 + 4980300 T^{2} + 23234867625062 T^{4} + 4980300 p^{8} T^{6} + p^{16} T^{8} \)
59$D_4\times C_2$ \( 1 - 26408088 T^{2} + 396550023159218 T^{4} - 26408088 p^{8} T^{6} + p^{16} T^{8} \)
61$D_{4}$ \( ( 1 + 10084 T + 53084286 T^{2} + 10084 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 792 T + 37671218 T^{2} + 792 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 71497380 T^{2} + 2526354220369862 T^{4} - 71497380 p^{8} T^{6} + p^{16} T^{8} \)
73$D_{4}$ \( ( 1 - 44 T + 42825726 T^{2} - 44 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 8868 T + 95944478 T^{2} - 8868 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 181435060 T^{2} + 12718407730707942 T^{4} - 181435060 p^{8} T^{6} + p^{16} T^{8} \)
89$D_4\times C_2$ \( 1 - 97686688 T^{2} + 9800859843406338 T^{4} - 97686688 p^{8} T^{6} + p^{16} T^{8} \)
97$D_{4}$ \( ( 1 - 13708 T + 217279038 T^{2} - 13708 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
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

−10.84803695825375675892535573059, −10.80693544336201295475119994632, −10.77058423902345769162767995327, −10.59240667688862522636706718877, −9.724306903103296351305786888932, −9.300108471976717138844016641575, −8.778761291732889550317254180504, −8.776338765626702950352932583335, −8.691358673445755338811491292137, −8.101466644287967689533201571444, −7.64198423100703328971210906800, −7.34236257637943697151803498539, −6.52235222721893519276187410121, −6.39558972005466015303304961855, −6.17660358491420999031001506089, −6.04914718707421195970427716632, −5.83689647519327057887894640108, −4.60624755953128935111878367634, −4.19251115211408132906753895843, −3.74349138203140334101296640408, −3.59308342954941308091646811226, −2.77067047971433933354780975817, −1.92090961219450480885373930013, −1.70368117018456975846546639600, −0.56448035647332114981281378946, 0.56448035647332114981281378946, 1.70368117018456975846546639600, 1.92090961219450480885373930013, 2.77067047971433933354780975817, 3.59308342954941308091646811226, 3.74349138203140334101296640408, 4.19251115211408132906753895843, 4.60624755953128935111878367634, 5.83689647519327057887894640108, 6.04914718707421195970427716632, 6.17660358491420999031001506089, 6.39558972005466015303304961855, 6.52235222721893519276187410121, 7.34236257637943697151803498539, 7.64198423100703328971210906800, 8.101466644287967689533201571444, 8.691358673445755338811491292137, 8.776338765626702950352932583335, 8.778761291732889550317254180504, 9.300108471976717138844016641575, 9.724306903103296351305786888932, 10.59240667688862522636706718877, 10.77058423902345769162767995327, 10.80693544336201295475119994632, 10.84803695825375675892535573059

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