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Properties

Label 8-363e4-1.1-c3e4-0-4
Degree $8$
Conductor $17363069361$
Sign $1$
Analytic cond. $210421.$
Root an. cond. $4.62792$
Motivic weight $3$
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  + 3·2-s + 12·3-s − 13·4-s + 12·5-s + 36·6-s + 11·7-s − 42·8-s + 90·9-s + 36·10-s − 156·12-s + 182·13-s + 33·14-s + 144·15-s + 126·16-s + 57·17-s + 270·18-s + 173·19-s − 156·20-s + 132·21-s − 42·23-s − 504·24-s − 271·25-s + 546·26-s + 540·27-s − 143·28-s + 651·29-s + 432·30-s + ⋯
L(s)  = 1  + 1.06·2-s + 2.30·3-s − 1.62·4-s + 1.07·5-s + 2.44·6-s + 0.593·7-s − 1.85·8-s + 10/3·9-s + 1.13·10-s − 3.75·12-s + 3.88·13-s + 0.629·14-s + 2.47·15-s + 1.96·16-s + 0.813·17-s + 3.53·18-s + 2.08·19-s − 1.74·20-s + 1.37·21-s − 0.380·23-s − 4.28·24-s − 2.16·25-s + 4.11·26-s + 3.84·27-s − 0.965·28-s + 4.16·29-s + 2.62·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(210421.\)
Root analytic conductor: \(4.62792\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 11^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(40.11578819\)
\(L(\frac12)\) \(\approx\) \(40.11578819\)
\(L(\frac{5}{2})\) 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)$
bad3$C_1$ \( ( 1 - p T )^{4} \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 11 p T^{2} - 63 T^{3} + 223 T^{4} - 63 p^{3} T^{5} + 11 p^{7} T^{6} - 3 p^{9} T^{7} + p^{12} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 83 p T^{2} - 2994 T^{3} + 67819 T^{4} - 2994 p^{3} T^{5} + 83 p^{7} T^{6} - 12 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 402 T^{2} - 946 p T^{3} + 162773 T^{4} - 946 p^{4} T^{5} + 402 p^{6} T^{6} - 11 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 14 p T + 16551 T^{2} - 1015624 T^{3} + 50687192 T^{4} - 1015624 p^{3} T^{5} + 16551 p^{6} T^{6} - 14 p^{10} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 57 T + 16021 T^{2} - 707631 T^{3} + 113209144 T^{4} - 707631 p^{3} T^{5} + 16021 p^{6} T^{6} - 57 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 173 T + 24849 T^{2} - 2173159 T^{3} + 197856056 T^{4} - 2173159 p^{3} T^{5} + 24849 p^{6} T^{6} - 173 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 42 T + 28427 T^{2} - 413280 T^{3} + 349199196 T^{4} - 413280 p^{3} T^{5} + 28427 p^{6} T^{6} + 42 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 651 T + 250093 T^{2} - 62670981 T^{3} + 11524701508 T^{4} - 62670981 p^{3} T^{5} + 250093 p^{6} T^{6} - 651 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 170 T + 122685 T^{2} + 14936530 T^{3} + 5536501547 T^{4} + 14936530 p^{3} T^{5} + 122685 p^{6} T^{6} + 170 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 244 T + 107679 T^{2} - 22139072 T^{3} + 5635850852 T^{4} - 22139072 p^{3} T^{5} + 107679 p^{6} T^{6} - 244 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 102 T + 111067 T^{2} - 1992132 T^{3} + 5701306888 T^{4} - 1992132 p^{3} T^{5} + 111067 p^{6} T^{6} + 102 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 322 T + 181243 T^{2} - 48993496 T^{3} + 21697488148 T^{4} - 48993496 p^{3} T^{5} + 181243 p^{6} T^{6} - 322 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 633 T + 461545 T^{2} + 166763625 T^{3} + 69749378908 T^{4} + 166763625 p^{3} T^{5} + 461545 p^{6} T^{6} + 633 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 468 T + 605161 T^{2} + 196671456 T^{3} + 135334830097 T^{4} + 196671456 p^{3} T^{5} + 605161 p^{6} T^{6} + 468 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 996 T + 928327 T^{2} + 561140838 T^{3} + 297148831615 T^{4} + 561140838 p^{3} T^{5} + 928327 p^{6} T^{6} + 996 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 413 T + 814107 T^{2} - 251879257 T^{3} + 270541264808 T^{4} - 251879257 p^{3} T^{5} + 814107 p^{6} T^{6} - 413 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 259 T + 1117027 T^{2} + 221089363 T^{3} + 492802725568 T^{4} + 221089363 p^{3} T^{5} + 1117027 p^{6} T^{6} + 259 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 237 T + 153589 T^{2} - 181426365 T^{3} + 211089307840 T^{4} - 181426365 p^{3} T^{5} + 153589 p^{6} T^{6} - 237 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 2309 T + 3335073 T^{2} - 44161957 p T^{3} + 2344749952988 T^{4} - 44161957 p^{4} T^{5} + 3335073 p^{6} T^{6} - 2309 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 2045 T + 3122832 T^{2} - 3169328350 T^{3} + 2582596622363 T^{4} - 3169328350 p^{3} T^{5} + 3122832 p^{6} T^{6} - 2045 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 639 T + 1399570 T^{2} - 960059370 T^{3} + 1037473710253 T^{4} - 960059370 p^{3} T^{5} + 1399570 p^{6} T^{6} - 639 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 894 T + 1324211 T^{2} - 258930180 T^{3} + 627382282200 T^{4} - 258930180 p^{3} T^{5} + 1324211 p^{6} T^{6} - 894 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 1432 T + 4102131 T^{2} - 3836622086 T^{3} + 5790534104459 T^{4} - 3836622086 p^{3} T^{5} + 4102131 p^{6} T^{6} - 1432 p^{9} T^{7} + p^{12} T^{8} \)
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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

−8.070654230529379970467777288287, −7.79865570862352087625044794432, −7.62390232639048481386205981246, −7.27858865444645429141951335638, −6.66722154687300635225920303453, −6.24281854319241990076992748303, −6.21979967884352463249839032847, −6.09945138863546661922629685648, −5.93669339248110258744768116442, −5.03832876072512723302673492736, −5.01745438354740539634657662927, −5.00586646578054182654963963910, −4.83692642054695395140633799037, −4.01157971325260375889435626280, −3.96982867188781134570203522897, −3.74653039255912445172128629095, −3.63905777768622434822364179198, −3.20404728300635448204219457517, −3.10362082609681586147673363744, −2.58591748960151859596904226079, −1.97931791964572434379445297650, −1.68038188609264105431953435119, −1.25340883988812275556306303640, −1.06847790642046712878792033272, −0.71547190815643770481459509126, 0.71547190815643770481459509126, 1.06847790642046712878792033272, 1.25340883988812275556306303640, 1.68038188609264105431953435119, 1.97931791964572434379445297650, 2.58591748960151859596904226079, 3.10362082609681586147673363744, 3.20404728300635448204219457517, 3.63905777768622434822364179198, 3.74653039255912445172128629095, 3.96982867188781134570203522897, 4.01157971325260375889435626280, 4.83692642054695395140633799037, 5.00586646578054182654963963910, 5.01745438354740539634657662927, 5.03832876072512723302673492736, 5.93669339248110258744768116442, 6.09945138863546661922629685648, 6.21979967884352463249839032847, 6.24281854319241990076992748303, 6.66722154687300635225920303453, 7.27858865444645429141951335638, 7.62390232639048481386205981246, 7.79865570862352087625044794432, 8.070654230529379970467777288287

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