Properties

Label 8-50e8-1.1-c1e4-0-0
Degree $8$
Conductor $3.906\times 10^{13}$
Sign $1$
Analytic cond. $158806.$
Root an. cond. $4.46795$
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  + 9·9-s + 10·11-s + 6·19-s − 4·29-s − 16·31-s + 24·41-s + 25·49-s + 32·59-s + 18·61-s + 2·71-s − 52·79-s + 44·81-s − 26·89-s + 90·99-s − 16·101-s + 22·109-s + 21·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 34·169-s + ⋯
L(s)  = 1  + 3·9-s + 3.01·11-s + 1.37·19-s − 0.742·29-s − 2.87·31-s + 3.74·41-s + 25/7·49-s + 4.16·59-s + 2.30·61-s + 0.237·71-s − 5.85·79-s + 44/9·81-s − 2.75·89-s + 9.04·99-s − 1.59·101-s + 2.10·109-s + 1.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.61·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(158806.\)
Root analytic conductor: \(4.46795\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2500} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{16} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(13.31290143\)
\(L(\frac12)\) \(\approx\) \(13.31290143\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$D_4\times C_2$ \( 1 - p^{2} T^{2} + 37 T^{4} - p^{4} T^{6} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 - 25 T^{2} + 253 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 26 T^{2} + 667 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 - 3 T + 39 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 85 T^{2} + 2853 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 90 T^{2} + 4043 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 41 T^{2} + 1837 T^{4} - 41 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 16 T + 3 p T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 9 T + 81 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 10 T^{2} + 7003 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - T + 111 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 229 T^{2} + 23617 T^{4} - 229 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2$ \( ( 1 + 13 T + p T^{2} )^{4} \)
83$D_4\times C_2$ \( 1 - 140 T^{2} + 13558 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 13 T + 219 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 115 T^{2} + 11313 T^{4} + 115 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.53603946395755069195005591377, −5.96230717351104819861041500264, −5.79513246825269741848861240989, −5.77183554536342898330701651165, −5.64415778349626668303193396116, −5.29528000484163959099731291304, −5.20063399667294723698147902263, −4.73653743544316360599099140270, −4.63396862927757360501259882633, −4.06820018560608554340896001726, −4.06418643539251438830989847431, −3.96994804323668625536757022247, −3.90539073371154426215315717753, −3.78976005464349541817415019704, −3.75174643198351229914560768422, −2.90926421208732648201384177879, −2.79891927563277161356798115452, −2.51910115788965853053624621621, −2.29153817861379380726177838638, −1.75736265569842592113232881297, −1.63699084149871091685059296801, −1.35115458686132719875381952762, −1.22875561702032368554407319576, −0.813357452381791073417815152508, −0.63168264772818072014805149516, 0.63168264772818072014805149516, 0.813357452381791073417815152508, 1.22875561702032368554407319576, 1.35115458686132719875381952762, 1.63699084149871091685059296801, 1.75736265569842592113232881297, 2.29153817861379380726177838638, 2.51910115788965853053624621621, 2.79891927563277161356798115452, 2.90926421208732648201384177879, 3.75174643198351229914560768422, 3.78976005464349541817415019704, 3.90539073371154426215315717753, 3.96994804323668625536757022247, 4.06418643539251438830989847431, 4.06820018560608554340896001726, 4.63396862927757360501259882633, 4.73653743544316360599099140270, 5.20063399667294723698147902263, 5.29528000484163959099731291304, 5.64415778349626668303193396116, 5.77183554536342898330701651165, 5.79513246825269741848861240989, 5.96230717351104819861041500264, 6.53603946395755069195005591377

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