Properties

Degree $8$
Conductor $41006250000$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 5·5-s + 8·7-s + 5·10-s + 2·11-s − 6·13-s − 8·14-s + 12·17-s − 2·22-s + 6·23-s + 10·25-s + 6·26-s − 12·31-s + 32-s − 12·34-s − 40·35-s + 13·37-s + 12·41-s + 4·43-s − 6·46-s + 2·47-s + 12·49-s − 10·50-s + 21·53-s − 10·55-s − 20·59-s − 2·61-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.23·5-s + 3.02·7-s + 1.58·10-s + 0.603·11-s − 1.66·13-s − 2.13·14-s + 2.91·17-s − 0.426·22-s + 1.25·23-s + 2·25-s + 1.17·26-s − 2.15·31-s + 0.176·32-s − 2.05·34-s − 6.76·35-s + 2.13·37-s + 1.87·41-s + 0.609·43-s − 0.884·46-s + 0.291·47-s + 12/7·49-s − 1.41·50-s + 2.88·53-s − 1.34·55-s − 2.60·59-s − 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 5^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{450} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.59290\)
\(L(\frac12)\) \(\approx\) \(1.59290\)
\(L(\frac{3}{2})\) 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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
3 \( 1 \)
5$C_4$ \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
11$C_4\times C_2$ \( 1 - 2 T + 13 T^{2} - 34 T^{3} + 225 T^{4} - 34 p T^{5} + 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2:C_4$ \( 1 + 6 T + 23 T^{2} + 120 T^{3} + 601 T^{4} + 120 p T^{5} + 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2:C_4$ \( 1 - 12 T + 37 T^{2} + 180 T^{3} - 1619 T^{4} + 180 p T^{5} + 37 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2:C_4$ \( 1 + 21 T^{2} - 10 T^{3} + 381 T^{4} - 10 p T^{5} + 21 p^{2} T^{6} + p^{4} T^{8} \)
23$C_4\times C_2$ \( 1 - 6 T + 13 T^{2} + 60 T^{3} - 659 T^{4} + 60 p T^{5} + 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2:C_4$ \( 1 - 19 T^{2} - 120 T^{3} + 721 T^{4} - 120 p T^{5} - 19 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2^2:C_4$ \( 1 + 12 T + 113 T^{2} + 834 T^{3} + 5605 T^{4} + 834 p T^{5} + 113 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2:C_4$ \( 1 - 13 T + 27 T^{2} + 445 T^{3} - 4264 T^{4} + 445 p T^{5} + 27 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2:C_4$ \( 1 - 12 T + 53 T^{2} - 444 T^{3} + 4405 T^{4} - 444 p T^{5} + 53 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2:C_4$ \( 1 - 2 T + 17 T^{2} + 130 T^{3} + 761 T^{4} + 130 p T^{5} + 17 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2:C_4$ \( 1 - 21 T + 253 T^{2} - 2535 T^{3} + 21196 T^{4} - 2535 p T^{5} + 253 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
59$C_4\times C_2$ \( 1 + 20 T + 181 T^{2} + 1600 T^{3} + 14601 T^{4} + 1600 p T^{5} + 181 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^2:C_4$ \( 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 424 p T^{5} + 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2^2:C_4$ \( 1 - 18 T + 77 T^{2} - 30 T^{3} + 961 T^{4} - 30 p T^{5} + 77 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
71$C_4\times C_2$ \( 1 + 28 T + 313 T^{2} + 2126 T^{3} + 14805 T^{4} + 2126 p T^{5} + 313 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^2:C_4$ \( 1 - 14 T + 3 T^{2} + 980 T^{3} - 9259 T^{4} + 980 p T^{5} + 3 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
79$C_4\times C_2$ \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \)
83$C_4\times C_2$ \( 1 - 6 T - 47 T^{2} + 780 T^{3} - 779 T^{4} + 780 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2^2:C_4$ \( 1 - 5 T - 79 T^{2} + 5 p T^{3} + 5276 T^{4} + 5 p^{2} T^{5} - 79 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^2:C_4$ \( 1 - 18 T + 47 T^{2} + 1740 T^{3} - 27059 T^{4} + 1740 p T^{5} + 47 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.977694205447265737245026042672, −7.66294323268935259553996609597, −7.61423358116035931439588100382, −7.54302845413740753981934791856, −7.48973596222358083568363141569, −6.91638770031444455740452687596, −6.90016401126305450011119505837, −6.33696216486950193548422891042, −5.86524893538001477862829959635, −5.62999756074611182535050525585, −5.51902732323806502111097695655, −5.04594811785913824421766271216, −5.04210035629149781568732689764, −4.51231501264071917048327503538, −4.48712796213655218263172040301, −4.19216809407218172486966430316, −3.89923457097070615483004516330, −3.56249681580468077846087389582, −3.07079600558539847574732498628, −2.97685753764307746585473005076, −2.43438669948652211811061482876, −1.73386855585515372242556521952, −1.70239122074357906676292239358, −0.896885527656177482216634935856, −0.71984320839594658026703611666, 0.71984320839594658026703611666, 0.896885527656177482216634935856, 1.70239122074357906676292239358, 1.73386855585515372242556521952, 2.43438669948652211811061482876, 2.97685753764307746585473005076, 3.07079600558539847574732498628, 3.56249681580468077846087389582, 3.89923457097070615483004516330, 4.19216809407218172486966430316, 4.48712796213655218263172040301, 4.51231501264071917048327503538, 5.04210035629149781568732689764, 5.04594811785913824421766271216, 5.51902732323806502111097695655, 5.62999756074611182535050525585, 5.86524893538001477862829959635, 6.33696216486950193548422891042, 6.90016401126305450011119505837, 6.91638770031444455740452687596, 7.48973596222358083568363141569, 7.54302845413740753981934791856, 7.61423358116035931439588100382, 7.66294323268935259553996609597, 7.977694205447265737245026042672

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