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Properties

Label 8-1449e4-1.1-c1e4-0-5
Degree $8$
Conductor $4.408\times 10^{12}$
Sign $1$
Analytic cond. $17921.8$
Root an. cond. $3.40151$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4-s − 2·5-s + 4·7-s + 2·8-s + 4·10-s − 8·11-s − 2·13-s − 8·14-s + 8·16-s − 8·19-s + 2·20-s + 16·22-s + 4·23-s − 11·25-s + 4·26-s − 4·28-s − 8·29-s + 4·31-s − 10·32-s − 8·35-s − 4·37-s + 16·38-s − 4·40-s − 12·41-s − 2·43-s + 8·44-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s − 0.894·5-s + 1.51·7-s + 0.707·8-s + 1.26·10-s − 2.41·11-s − 0.554·13-s − 2.13·14-s + 2·16-s − 1.83·19-s + 0.447·20-s + 3.41·22-s + 0.834·23-s − 2.19·25-s + 0.784·26-s − 0.755·28-s − 1.48·29-s + 0.718·31-s − 1.76·32-s − 1.35·35-s − 0.657·37-s + 2.59·38-s − 0.632·40-s − 1.87·41-s − 0.304·43-s + 1.20·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 7^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(17921.8\)
Root analytic conductor: \(3.40151\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{8} \cdot 7^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7$C_1$ \( ( 1 - T )^{4} \)
23$C_1$ \( ( 1 - T )^{4} \)
good2$C_2 \wr C_2\wr C_2$ \( 1 + p T + 5 T^{2} + 5 p T^{3} + 13 T^{4} + 5 p^{2} T^{5} + 5 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 3 p T^{2} + 24 T^{3} + 101 T^{4} + 24 p T^{5} + 3 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 52 T^{2} + 20 p T^{3} + 829 T^{4} + 20 p^{2} T^{5} + 52 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 17 T^{2} - 8 T^{3} + 55 T^{4} - 8 p T^{5} + 17 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 50 T^{2} - 16 T^{3} + 67 p T^{4} - 16 p T^{5} + 50 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 70 T^{2} + 336 T^{3} + 1763 T^{4} + 336 p T^{5} + 70 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 110 T^{2} + 640 T^{3} + 4651 T^{4} + 640 p T^{5} + 110 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 52 T^{2} - 420 T^{3} + 1421 T^{4} - 420 p T^{5} + 52 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 20 T^{2} - 180 T^{3} - 1755 T^{4} - 180 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 84 T^{2} + 604 T^{3} + 4757 T^{4} + 604 p T^{5} + 84 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 131 T^{2} + 264 T^{3} + 7791 T^{4} + 264 p T^{5} + 131 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 156 T^{2} + 1232 T^{3} + 9222 T^{4} + 1232 p T^{5} + 156 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 135 T^{2} - 272 T^{3} + 9899 T^{4} - 272 p T^{5} + 135 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 22 T + 237 T^{2} + 1544 T^{3} + 9747 T^{4} + 1544 p T^{5} + 237 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 22 T + 333 T^{2} - 3320 T^{3} + 29001 T^{4} - 3320 p T^{5} + 333 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 225 T^{2} + 1796 T^{3} + 21059 T^{4} + 1796 p T^{5} + 225 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 46 T + 1029 T^{2} + 14628 T^{3} + 145493 T^{4} + 14628 p T^{5} + 1029 p^{2} T^{6} + 46 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 316 T^{2} + 3284 T^{3} + 35053 T^{4} + 3284 p T^{5} + 316 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 196 T^{2} - 1460 T^{3} + 15037 T^{4} - 1460 p T^{5} + 196 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 90 T^{2} + 1032 T^{3} - 2269 T^{4} + 1032 p T^{5} + 90 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 34 T + 671 T^{2} + 9272 T^{3} + 99277 T^{4} + 9272 p T^{5} + 671 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 110 T^{2} - 1408 T^{3} - 9381 T^{4} - 1408 p T^{5} + 110 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.51039557222115745086939640217, −7.19768271770094931055091007013, −6.89530605928380126533230829220, −6.71564651288330159033505106747, −6.38348899878331608097486707110, −6.08267645739920930578407308236, −5.83098624788739545112964869548, −5.55390386843500072023791192832, −5.54923108233996494949466598593, −5.31248194038059422200757493095, −4.95350728577447142796534848196, −4.89520264559482491669912567369, −4.56893867806062625997458280930, −4.28845297771897567160803456640, −4.18254929515750532470716379538, −3.97156600183317253188879815063, −3.53434126585843400620190006400, −3.28622703884232841217949607852, −3.06198789609993881235584252469, −2.77367237532757570553731331136, −2.39953716149398114765788171161, −2.24043287633578655260408926465, −1.62846200567657853501367336518, −1.47786163702019738409619452006, −1.32072307313678524800888992946, 0, 0, 0, 0, 1.32072307313678524800888992946, 1.47786163702019738409619452006, 1.62846200567657853501367336518, 2.24043287633578655260408926465, 2.39953716149398114765788171161, 2.77367237532757570553731331136, 3.06198789609993881235584252469, 3.28622703884232841217949607852, 3.53434126585843400620190006400, 3.97156600183317253188879815063, 4.18254929515750532470716379538, 4.28845297771897567160803456640, 4.56893867806062625997458280930, 4.89520264559482491669912567369, 4.95350728577447142796534848196, 5.31248194038059422200757493095, 5.54923108233996494949466598593, 5.55390386843500072023791192832, 5.83098624788739545112964869548, 6.08267645739920930578407308236, 6.38348899878331608097486707110, 6.71564651288330159033505106747, 6.89530605928380126533230829220, 7.19768271770094931055091007013, 7.51039557222115745086939640217

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