Properties

Label 8-12e12-1.1-c1e4-0-3
Degree $8$
Conductor $8.916\times 10^{12}$
Sign $1$
Analytic cond. $36247.9$
Root an. cond. $3.71458$
Motivic weight $1$
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  + 4·5-s − 12·7-s − 2·11-s + 8·13-s − 16·17-s + 12·19-s − 12·23-s + 11·25-s + 8·31-s − 48·35-s − 24·37-s + 18·41-s − 14·43-s + 8·47-s + 71·49-s − 16·53-s − 8·55-s − 6·59-s − 12·61-s + 32·65-s + 2·67-s + 24·77-s − 22·83-s − 64·85-s − 96·91-s + 48·95-s − 2·97-s + ⋯
L(s)  = 1  + 1.78·5-s − 4.53·7-s − 0.603·11-s + 2.21·13-s − 3.88·17-s + 2.75·19-s − 2.50·23-s + 11/5·25-s + 1.43·31-s − 8.11·35-s − 3.94·37-s + 2.81·41-s − 2.13·43-s + 1.16·47-s + 71/7·49-s − 2.19·53-s − 1.07·55-s − 0.781·59-s − 1.53·61-s + 3.96·65-s + 0.244·67-s + 2.73·77-s − 2.41·83-s − 6.94·85-s − 10.0·91-s + 4.92·95-s − 0.203·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(36247.9\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{12} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4740836890\)
\(L(\frac12)\) \(\approx\) \(0.4740836890\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$D_4\times C_2$ \( 1 - 4 T + p T^{2} + 8 T^{3} - 44 T^{4} + 8 p T^{5} + p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 + 12 T + 73 T^{2} + 300 T^{3} + 912 T^{4} + 300 p T^{5} + 73 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 + 2 T + 5 T^{2} + 26 T^{3} - 32 T^{4} + 26 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2$$\times$$C_2^2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 12 T + 97 T^{2} + 588 T^{3} + 2976 T^{4} + 588 p T^{5} + 97 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 + 9 T^{2} + 156 T^{3} - 484 T^{4} + 156 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 8 T + 13 T^{2} + 88 T^{3} - 344 T^{4} + 88 p T^{5} + 13 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 24 T + 288 T^{2} + 2472 T^{3} + 16862 T^{4} + 2472 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 18 T + 181 T^{2} - 1314 T^{3} + 8076 T^{4} - 1314 p T^{5} + 181 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 14 T + 65 T^{2} - 474 T^{3} - 6280 T^{4} - 474 p T^{5} + 65 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 8 T - 19 T^{2} + 88 T^{3} + 1672 T^{4} + 88 p T^{5} - 19 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 16 T + 128 T^{2} + 1264 T^{3} + 11806 T^{4} + 1264 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 6 T + 45 T^{2} + 462 T^{3} + 848 T^{4} + 462 p T^{5} + 45 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 12 T + 261 T^{2} + 2088 T^{3} + 24452 T^{4} + 2088 p T^{5} + 261 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 2 T + 65 T^{2} - 762 T^{3} + 2600 T^{4} - 762 p T^{5} + 65 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 236 T^{2} + 23814 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^3$ \( 1 - 155 T^{2} + 17784 T^{4} - 155 p^{2} T^{6} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 + 22 T + 185 T^{2} - 290 T^{3} - 14024 T^{4} - 290 p T^{5} + 185 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 + 36 T^{2} + 13094 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.45511842571102296900110495765, −6.37982838125132915661697156251, −6.29852304518973334001398976989, −5.97293396243320623592485619866, −5.87848156989204378673312811230, −5.86331573223766769079580336707, −5.65459380116066627054827743565, −5.10740455173611621331015768599, −5.00802930288598194060794768433, −4.60486895372323910836754502598, −4.55308548518387847710116764308, −3.98038449687516917327486690671, −3.94273074779008143021655469915, −3.60122844268871607785424687077, −3.43460322865860853170710630332, −3.25326565838557088583836862069, −2.82016777546869339458220504407, −2.77412209660101194973492465350, −2.74070165851088063151620858803, −2.12920109650909918470528262684, −1.93922166922713445916469938068, −1.50295639127559131735070041286, −1.32650019315594894267096988009, −0.49899007974051927870410641547, −0.19247719208642159013192524329, 0.19247719208642159013192524329, 0.49899007974051927870410641547, 1.32650019315594894267096988009, 1.50295639127559131735070041286, 1.93922166922713445916469938068, 2.12920109650909918470528262684, 2.74070165851088063151620858803, 2.77412209660101194973492465350, 2.82016777546869339458220504407, 3.25326565838557088583836862069, 3.43460322865860853170710630332, 3.60122844268871607785424687077, 3.94273074779008143021655469915, 3.98038449687516917327486690671, 4.55308548518387847710116764308, 4.60486895372323910836754502598, 5.00802930288598194060794768433, 5.10740455173611621331015768599, 5.65459380116066627054827743565, 5.86331573223766769079580336707, 5.87848156989204378673312811230, 5.97293396243320623592485619866, 6.29852304518973334001398976989, 6.37982838125132915661697156251, 6.45511842571102296900110495765

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