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Properties

Label 8-98e4-1.1-c2e4-0-0
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  + 4·4-s + 18·9-s − 36·11-s + 12·16-s − 60·23-s + 46·25-s + 48·29-s + 72·36-s + 124·37-s − 8·43-s − 144·44-s + 156·53-s + 32·64-s + 116·67-s − 24·71-s − 220·79-s + 153·81-s − 240·92-s − 648·99-s + 184·100-s − 132·107-s − 140·109-s + 240·113-s + 192·116-s + 362·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 4-s + 2·9-s − 3.27·11-s + 3/4·16-s − 2.60·23-s + 1.83·25-s + 1.65·29-s + 2·36-s + 3.35·37-s − 0.186·43-s − 3.27·44-s + 2.94·53-s + 1/2·64-s + 1.73·67-s − 0.338·71-s − 2.78·79-s + 17/9·81-s − 2.60·92-s − 6.54·99-s + 1.83·100-s − 1.23·107-s − 1.28·109-s + 2.12·113-s + 1.65·116-s + 2.99·121-s + 0.00787·127-s + 0.00763·131-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

\(L(\frac{3}{2})\) \(\approx\) \(2.571122128\)
\(L(\frac12)\) \(\approx\) \(2.571122128\)
\(L(2)\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
7 \( 1 \)
good3$D_4\times C_2$ \( 1 - 2 p^{2} T^{2} + 19 p^{2} T^{4} - 2 p^{6} T^{6} + p^{8} T^{8} \)
5$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T - 21 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )( 1 + 2 T - 21 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} ) \)
11$D_{4}$ \( ( 1 + 18 T + 305 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 412 T^{2} + 89190 T^{4} - 412 p^{4} T^{6} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 - 958 T^{2} + 1347 p^{2} T^{4} - 958 p^{4} T^{6} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 - 1426 T^{2} + 768939 T^{4} - 1426 p^{4} T^{6} + p^{8} T^{8} \)
23$D_{4}$ \( ( 1 + 30 T + 1121 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 - 24 T + 1754 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 850 T^{2} + 1233867 T^{4} - 850 p^{4} T^{6} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 - 62 T + 2547 T^{2} - 62 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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$D_4\times C_2$ \( 1 - 3778 T^{2} + 13267131 T^{4} - 3778 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 - 78 T + 6851 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 5410 T^{2} + 23946747 T^{4} - 5410 p^{4} T^{6} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 - 2302 T^{2} + 25403811 T^{4} - 2302 p^{4} T^{6} + p^{8} T^{8} \)
67$D_{4}$ \( ( 1 - 58 T + 5769 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 + 12 T + 8318 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 1390 T^{2} + 5504019 T^{4} - 1390 p^{4} T^{6} + p^{8} T^{8} \)
79$D_{4}$ \( ( 1 + 110 T + 15057 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 380 T^{2} + 89625894 T^{4} + 380 p^{4} T^{6} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 958 T^{2} - 38888445 T^{4} - 958 p^{4} T^{6} + p^{8} T^{8} \)
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

−9.984984765823436223573531734614, −9.963814611820126669969414253910, −9.936923525599836007299780831419, −9.138492025446357297643928110634, −8.838948530614137720132247608758, −8.520707584269449788381752231035, −7.894663502334751161153183135360, −7.87603456012005304510217729510, −7.76572456155644459538457214736, −7.59466875598568499696646299045, −6.89902441258110343322973441282, −6.72674837061930701824794082812, −6.55976036099246848100284737732, −6.01535387850148020994597155367, −5.63115656179015135495985012503, −5.26603376121205831469878018386, −5.08541992710535575449551569317, −4.45550139028975736656622976064, −4.08898896463968764285484657700, −4.00685145887804941586479636368, −2.84823006573045957364810800657, −2.64393057434298362704356319025, −2.53023533207106782689177136474, −1.67968749432020468768486290086, −0.78719332633456957535995776613, 0.78719332633456957535995776613, 1.67968749432020468768486290086, 2.53023533207106782689177136474, 2.64393057434298362704356319025, 2.84823006573045957364810800657, 4.00685145887804941586479636368, 4.08898896463968764285484657700, 4.45550139028975736656622976064, 5.08541992710535575449551569317, 5.26603376121205831469878018386, 5.63115656179015135495985012503, 6.01535387850148020994597155367, 6.55976036099246848100284737732, 6.72674837061930701824794082812, 6.89902441258110343322973441282, 7.59466875598568499696646299045, 7.76572456155644459538457214736, 7.87603456012005304510217729510, 7.894663502334751161153183135360, 8.520707584269449788381752231035, 8.838948530614137720132247608758, 9.138492025446357297643928110634, 9.936923525599836007299780831419, 9.963814611820126669969414253910, 9.984984765823436223573531734614

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