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Properties

Label 8-304e4-1.1-c2e4-0-0
Degree $8$
Conductor $8540717056$
Sign $1$
Analytic cond. $4707.97$
Root an. cond. $2.87808$
Motivic weight $2$
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  − 6·3-s + 2·5-s − 8·7-s + 5·9-s + 40·11-s − 30·13-s − 12·15-s + 14·17-s − 16·19-s + 48·21-s + 10·23-s + 51·25-s + 42·27-s + 66·29-s − 240·33-s − 16·35-s + 180·39-s + 18·41-s − 38·43-s + 10·45-s + 70·47-s − 108·49-s − 84·51-s − 42·53-s + 80·55-s + 96·57-s − 42·59-s + ⋯
L(s)  = 1  − 2·3-s + 2/5·5-s − 8/7·7-s + 5/9·9-s + 3.63·11-s − 2.30·13-s − 4/5·15-s + 0.823·17-s − 0.842·19-s + 16/7·21-s + 0.434·23-s + 2.03·25-s + 14/9·27-s + 2.27·29-s − 7.27·33-s − 0.457·35-s + 4.61·39-s + 0.439·41-s − 0.883·43-s + 2/9·45-s + 1.48·47-s − 2.20·49-s − 1.64·51-s − 0.792·53-s + 1.45·55-s + 1.68·57-s − 0.711·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(4707.97\)
Root analytic conductor: \(2.87808\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 19^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6806721391\)
\(L(\frac12)\) \(\approx\) \(0.6806721391\)
\(L(2)\) 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 \( 1 \)
19$C_2^2$ \( 1 + 16 T + 30 p T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} \)
good3$D_4\times C_2$ \( 1 + 2 p T + 31 T^{2} + 38 p T^{3} + 388 T^{4} + 38 p^{3} T^{5} + 31 p^{4} T^{6} + 2 p^{7} T^{7} + p^{8} T^{8} \)
5$C_2^2$ \( ( 1 - T - 24 T^{2} - p^{2} T^{3} + p^{4} T^{4} )^{2} \)
7$D_{4}$ \( ( 1 + 4 T + 78 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
11$D_{4}$ \( ( 1 - 20 T + 318 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + 30 T + 641 T^{2} + 10230 T^{3} + 138420 T^{4} + 10230 p^{2} T^{5} + 641 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 - 14 T - 407 T^{2} - 350 T^{3} + 223444 T^{4} - 350 p^{2} T^{5} - 407 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 - 10 T - 497 T^{2} + 4610 T^{3} + 23668 T^{4} + 4610 p^{2} T^{5} - 497 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} \)
29$D_4\times C_2$ \( 1 - 66 T + 2849 T^{2} - 92202 T^{3} + 2465460 T^{4} - 92202 p^{2} T^{5} + 2849 p^{4} T^{6} - 66 p^{6} T^{7} + p^{8} T^{8} \)
31$C_2^2$ \( ( 1 - 950 T^{2} + p^{4} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 4156 T^{2} + 8035302 T^{4} - 4156 p^{4} T^{6} + p^{8} T^{8} \)
41$D_4\times C_2$ \( 1 - 18 T - 31 T^{2} + 2502 T^{3} - 2624892 T^{4} + 2502 p^{2} T^{5} - 31 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \)
43$D_4\times C_2$ \( 1 + 38 T + 13 p T^{2} - 106894 T^{3} - 5305532 T^{4} - 106894 p^{2} T^{5} + 13 p^{5} T^{6} + 38 p^{6} T^{7} + p^{8} T^{8} \)
47$D_4\times C_2$ \( 1 - 70 T + 1423 T^{2} + 65870 T^{3} - 3614252 T^{4} + 65870 p^{2} T^{5} + 1423 p^{4} T^{6} - 70 p^{6} T^{7} + p^{8} T^{8} \)
53$D_4\times C_2$ \( 1 + 42 T + 1745 T^{2} + 48594 T^{3} - 4900140 T^{4} + 48594 p^{2} T^{5} + 1745 p^{4} T^{6} + 42 p^{6} T^{7} + p^{8} T^{8} \)
59$D_4\times C_2$ \( 1 + 42 T + 5519 T^{2} + 207102 T^{3} + 14244228 T^{4} + 207102 p^{2} T^{5} + 5519 p^{4} T^{6} + 42 p^{6} T^{7} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 - 74 T - 1799 T^{2} + 12358 T^{3} + 18703588 T^{4} + 12358 p^{2} T^{5} - 1799 p^{4} T^{6} - 74 p^{6} T^{7} + p^{8} T^{8} \)
67$D_4\times C_2$ \( 1 + 102 T + 5375 T^{2} + 194514 T^{3} - 946620 T^{4} + 194514 p^{2} T^{5} + 5375 p^{4} T^{6} + 102 p^{6} T^{7} + p^{8} T^{8} \)
71$D_4\times C_2$ \( 1 + 306 T + 48935 T^{2} + 5423238 T^{3} + 446032740 T^{4} + 5423238 p^{2} T^{5} + 48935 p^{4} T^{6} + 306 p^{6} T^{7} + p^{8} T^{8} \)
73$D_4\times C_2$ \( 1 - 98 T + 2689 T^{2} + 366814 T^{3} - 31760732 T^{4} + 366814 p^{2} T^{5} + 2689 p^{4} T^{6} - 98 p^{6} T^{7} + p^{8} T^{8} \)
79$D_4\times C_2$ \( 1 - 126 T + 18215 T^{2} - 1628298 T^{3} + 161081220 T^{4} - 1628298 p^{2} T^{5} + 18215 p^{4} T^{6} - 126 p^{6} T^{7} + p^{8} T^{8} \)
83$D_{4}$ \( ( 1 - 32 T + 13818 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 6 T + 15785 T^{2} + 94638 T^{3} + 186140340 T^{4} + 94638 p^{2} T^{5} + 15785 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 + 486 T + 117161 T^{2} + 18676494 T^{3} + 2129048148 T^{4} + 18676494 p^{2} T^{5} + 117161 p^{4} T^{6} + 486 p^{6} T^{7} + p^{8} T^{8} \)
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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

−8.617190867174253233123163185763, −8.014969722305128419791670282199, −7.61954888245875700108448328265, −7.40648978018492637906392433871, −7.01315247781292169415759799247, −6.74901431973253637646504825098, −6.58616266788696888014773460101, −6.40208390160890462718742213172, −6.35848588296310429413494098485, −6.11157304712838105644859695715, −5.65995738231737544707615930747, −5.32611783762862693041791771433, −5.22215289309745251418540249291, −4.74985215152928088381888849199, −4.62459706923003704702182167972, −4.16999249029470649433325052208, −4.11590298658617140626371755730, −3.34173305790262830514802792830, −3.18783036029967824064938705486, −2.69455622611018578918757335415, −2.68350368831210221323304851382, −1.62418567435985871527786141012, −1.41569306446485322127542838231, −0.819888945744019031560038321134, −0.30096569446447775367454939932, 0.30096569446447775367454939932, 0.819888945744019031560038321134, 1.41569306446485322127542838231, 1.62418567435985871527786141012, 2.68350368831210221323304851382, 2.69455622611018578918757335415, 3.18783036029967824064938705486, 3.34173305790262830514802792830, 4.11590298658617140626371755730, 4.16999249029470649433325052208, 4.62459706923003704702182167972, 4.74985215152928088381888849199, 5.22215289309745251418540249291, 5.32611783762862693041791771433, 5.65995738231737544707615930747, 6.11157304712838105644859695715, 6.35848588296310429413494098485, 6.40208390160890462718742213172, 6.58616266788696888014773460101, 6.74901431973253637646504825098, 7.01315247781292169415759799247, 7.40648978018492637906392433871, 7.61954888245875700108448328265, 8.014969722305128419791670282199, 8.617190867174253233123163185763

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