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Properties

Label 8-98e4-1.1-c2e4-0-1
Degree $8$
Conductor $92236816$
Sign $1$
Analytic cond. $50.8445$
Root an. cond. $1.63410$
Motivic weight $2$
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  + 6·3-s − 2·4-s + 6·5-s + 9·9-s + 18·11-s − 12·12-s + 36·15-s + 30·17-s − 6·19-s − 12·20-s + 30·23-s − 5·25-s − 18·27-s + 48·29-s + 42·31-s + 108·33-s − 18·36-s − 62·37-s − 8·43-s − 36·44-s + 54·45-s − 174·47-s + 180·51-s − 78·53-s + 108·55-s − 36·57-s + 78·59-s + ⋯
L(s)  = 1  + 2·3-s − 1/2·4-s + 6/5·5-s + 9-s + 1.63·11-s − 12-s + 12/5·15-s + 1.76·17-s − 0.315·19-s − 3/5·20-s + 1.30·23-s − 1/5·25-s − 2/3·27-s + 1.65·29-s + 1.35·31-s + 3.27·33-s − 1/2·36-s − 1.67·37-s − 0.186·43-s − 0.818·44-s + 6/5·45-s − 3.70·47-s + 3.52·51-s − 1.47·53-s + 1.96·55-s − 0.631·57-s + 1.32·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(92236816\)    =    \(2^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(50.8445\)
Root analytic conductor: \(1.63410\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 92236816,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.316065247\)
\(L(\frac12)\) \(\approx\) \(5.316065247\)
\(L(2)\) 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$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
7 \( 1 \)
good3$D_4\times C_2$ \( 1 - 2 p T + p^{3} T^{2} - 10 p^{2} T^{3} + 28 p^{2} T^{4} - 10 p^{4} T^{5} + p^{7} T^{6} - 2 p^{7} T^{7} + p^{8} T^{8} \)
5$C_2$$\times$$C_2^2$ \( ( 1 - 2 T + p^{2} T^{2} )^{2}( 1 - 2 T - 21 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} ) \)
11$D_4\times C_2$ \( 1 - 18 T + 19 T^{2} - 1134 T^{3} + 39180 T^{4} - 1134 p^{2} T^{5} + 19 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \)
13$D_4\times C_2$ \( 1 - 412 T^{2} + 89190 T^{4} - 412 p^{4} T^{6} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 - 30 T + 929 T^{2} - 1110 p T^{3} + 1380 p^{2} T^{4} - 1110 p^{3} T^{5} + 929 p^{4} T^{6} - 30 p^{6} T^{7} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 + 6 T + 731 T^{2} + 4314 T^{3} + 390972 T^{4} + 4314 p^{2} T^{5} + 731 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 - 30 T - 221 T^{2} - 1890 T^{3} + 500700 T^{4} - 1890 p^{2} T^{5} - 221 p^{4} T^{6} - 30 p^{6} T^{7} + p^{8} T^{8} \)
29$D_{4}$ \( ( 1 - 24 T + 1754 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 42 T + 1307 T^{2} - 30198 T^{3} + 158508 T^{4} - 30198 p^{2} T^{5} + 1307 p^{4} T^{6} - 42 p^{6} T^{7} + p^{8} T^{8} \)
37$D_4\times C_2$ \( 1 + 62 T + 1297 T^{2} - 11842 T^{3} - 649388 T^{4} - 11842 p^{2} T^{5} + 1297 p^{4} T^{6} + 62 p^{6} T^{7} + p^{8} T^{8} \)
41$D_4\times C_2$ \( 1 - 5500 T^{2} + 13185222 T^{4} - 5500 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 + 4 T + 3630 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 174 T + 17027 T^{2} + 1206690 T^{3} + 65507772 T^{4} + 1206690 p^{2} T^{5} + 17027 p^{4} T^{6} + 174 p^{6} T^{7} + p^{8} T^{8} \)
53$D_4\times C_2$ \( 1 + 78 T - 767 T^{2} + 96174 T^{3} + 21955764 T^{4} + 96174 p^{2} T^{5} - 767 p^{4} T^{6} + 78 p^{6} T^{7} + p^{8} T^{8} \)
59$D_4\times C_2$ \( 1 - 78 T + 5747 T^{2} - 290082 T^{3} + 8773068 T^{4} - 290082 p^{2} T^{5} + 5747 p^{4} T^{6} - 78 p^{6} T^{7} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 - 42 T + 2033 T^{2} - 60690 T^{3} - 9569868 T^{4} - 60690 p^{2} T^{5} + 2033 p^{4} T^{6} - 42 p^{6} T^{7} + p^{8} T^{8} \)
67$D_4\times C_2$ \( 1 + 58 T - 2405 T^{2} - 186122 T^{3} - 1970756 T^{4} - 186122 p^{2} T^{5} - 2405 p^{4} T^{6} + 58 p^{6} T^{7} + p^{8} T^{8} \)
71$D_{4}$ \( ( 1 + 12 T + 8318 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 318 T + 51257 T^{2} + 5580582 T^{3} + 459199092 T^{4} + 5580582 p^{2} T^{5} + 51257 p^{4} T^{6} + 318 p^{6} T^{7} + p^{8} T^{8} \)
79$D_4\times C_2$ \( 1 - 110 T - 2957 T^{2} - 283250 T^{3} + 112247068 T^{4} - 283250 p^{2} T^{5} - 2957 p^{4} T^{6} - 110 p^{6} T^{7} + p^{8} T^{8} \)
83$D_4\times C_2$ \( 1 + 380 T^{2} + 89625894 T^{4} + 380 p^{4} T^{6} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 378 T + 71921 T^{2} - 9182754 T^{3} + 904668996 T^{4} - 9182754 p^{2} T^{5} + 71921 p^{4} T^{6} - 378 p^{6} T^{7} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 - 26620 T^{2} + 330657414 T^{4} - 26620 p^{4} T^{6} + p^{8} T^{8} \)
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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

−10.04523456200348191704711402781, −9.669300091096214436293087544730, −9.248946217556932842823891177419, −9.222739081761717664485096981223, −8.999103561798808942099459505974, −8.483126808279993467300378711006, −8.397298115573614762390082374820, −8.168661632715784215042906577405, −7.953640889798224486903233493717, −7.40591811615045968164589285997, −6.96034227638810792194524600757, −6.79140965754722714968400936207, −6.31637125407610139709159408744, −6.09300253841617803824441159545, −5.77391738536912077315832754134, −5.20169040050743942140392144696, −4.75778701532085588957304940423, −4.68415662604099754773808647517, −3.99908205434079249642961652218, −3.28655698630612857261358113689, −3.17965751167893178565607529241, −3.14706550373047685138883108826, −2.24678407984407438558186999213, −1.71645028168781663764494516786, −1.14751311318133873880998113656, 1.14751311318133873880998113656, 1.71645028168781663764494516786, 2.24678407984407438558186999213, 3.14706550373047685138883108826, 3.17965751167893178565607529241, 3.28655698630612857261358113689, 3.99908205434079249642961652218, 4.68415662604099754773808647517, 4.75778701532085588957304940423, 5.20169040050743942140392144696, 5.77391738536912077315832754134, 6.09300253841617803824441159545, 6.31637125407610139709159408744, 6.79140965754722714968400936207, 6.96034227638810792194524600757, 7.40591811615045968164589285997, 7.953640889798224486903233493717, 8.168661632715784215042906577405, 8.397298115573614762390082374820, 8.483126808279993467300378711006, 8.999103561798808942099459505974, 9.222739081761717664485096981223, 9.248946217556932842823891177419, 9.669300091096214436293087544730, 10.04523456200348191704711402781

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