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Properties

Label 8-45e8-1.1-c1e4-0-8
Degree $8$
Conductor $1.682\times 10^{13}$
Sign $1$
Analytic cond. $68361.0$
Root an. cond. $4.02115$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 7-s − 8-s + 11-s − 2·13-s + 2·14-s + 16-s + 11·17-s + 2·19-s + 2·22-s + 15·23-s − 4·26-s − 29-s − 4·31-s − 2·32-s + 22·34-s + 37-s + 4·38-s + 5·41-s + 10·43-s + 30·46-s + 20·47-s − 15·49-s + 20·53-s − 56-s − 2·58-s − 17·59-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.377·7-s − 0.353·8-s + 0.301·11-s − 0.554·13-s + 0.534·14-s + 1/4·16-s + 2.66·17-s + 0.458·19-s + 0.426·22-s + 3.12·23-s − 0.784·26-s − 0.185·29-s − 0.718·31-s − 0.353·32-s + 3.77·34-s + 0.164·37-s + 0.648·38-s + 0.780·41-s + 1.52·43-s + 4.42·46-s + 2.91·47-s − 2.14·49-s + 2.74·53-s − 0.133·56-s − 0.262·58-s − 2.21·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{16} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(68361.0\)
Root analytic conductor: \(4.02115\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{16} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(14.29786961\)
\(L(\frac12)\) \(\approx\) \(14.29786961\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - p T + p^{2} T^{2} - 7 T^{3} + 11 T^{4} - 7 p T^{5} + p^{4} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - T + 16 T^{2} + 3 T^{3} + 117 T^{4} + 3 p T^{5} + 16 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - T + 19 T^{2} - 74 T^{3} + 167 T^{4} - 74 p T^{5} + 19 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 22 T^{2} + 73 T^{3} + 341 T^{4} + 73 p T^{5} + 22 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 88 T^{2} - 451 T^{3} + 2111 T^{4} - 451 p T^{5} + 88 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 49 T^{2} - 34 T^{3} + 1115 T^{4} - 34 p T^{5} + 49 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 15 T + 149 T^{2} - 1026 T^{3} + 5553 T^{4} - 1026 p T^{5} + 149 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + T + 76 T^{2} - 55 T^{3} + 2597 T^{4} - 55 p T^{5} + 76 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 82 T^{2} + 345 T^{3} + 3405 T^{4} + 345 p T^{5} + 82 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - T + 49 T^{2} + 392 T^{3} + 241 T^{4} + 392 p T^{5} + 49 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 139 T^{2} - 454 T^{3} + 7829 T^{4} - 454 p T^{5} + 139 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 124 T^{2} - 704 T^{3} + 6293 T^{4} - 704 p T^{5} + 124 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 295 T^{2} - 2890 T^{3} + 22931 T^{4} - 2890 p T^{5} + 295 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 298 T^{2} - 3001 T^{3} + 25499 T^{4} - 3001 p T^{5} + 298 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 17 T + 238 T^{2} + 2095 T^{3} + 18809 T^{4} + 2095 p T^{5} + 238 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 13 T + 241 T^{2} + 2288 T^{3} + 21959 T^{4} + 2288 p T^{5} + 241 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 17 T + 304 T^{2} + 3117 T^{3} + 32001 T^{4} + 3117 p T^{5} + 304 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 244 T^{2} - 1441 T^{3} + 24947 T^{4} - 1441 p T^{5} + 244 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 196 T^{2} + 679 T^{3} + 18071 T^{4} + 679 p T^{5} + 196 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 283 T^{2} + 1590 T^{3} + 32439 T^{4} + 1590 p T^{5} + 283 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 30 T + 620 T^{2} - 8415 T^{3} + 89871 T^{4} - 8415 p T^{5} + 620 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 257 T^{2} - 1998 T^{3} + 31929 T^{4} - 1998 p T^{5} + 257 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 19 T + 469 T^{2} - 5368 T^{3} + 71215 T^{4} - 5368 p T^{5} + 469 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.52305652331659294763790874880, −6.07534980089151574561857576228, −5.94612734004827364700429546065, −5.78076389487534389143791738107, −5.62127766350659225704815069152, −5.26723511887452455028540333603, −5.10827318917029329979696675003, −5.01806034025202991336412962576, −4.99589375078119865612437326590, −4.45685587889652391463976540480, −4.39356677334090133886398563412, −4.21455659274653560642816607563, −4.06510909967445675213997171068, −3.58278698217385449218391801847, −3.38136072386547024421688973230, −3.37199276414255575988308015546, −2.94734977746183342000913008259, −2.81529782874818295572378536885, −2.63775681685815934906943331248, −2.02435309273472873150061198243, −1.85029490493925111992155641512, −1.51297627303950775529521207966, −0.962506383000096193492673335512, −0.874156326520339318711466541554, −0.62711785145750483810458552161, 0.62711785145750483810458552161, 0.874156326520339318711466541554, 0.962506383000096193492673335512, 1.51297627303950775529521207966, 1.85029490493925111992155641512, 2.02435309273472873150061198243, 2.63775681685815934906943331248, 2.81529782874818295572378536885, 2.94734977746183342000913008259, 3.37199276414255575988308015546, 3.38136072386547024421688973230, 3.58278698217385449218391801847, 4.06510909967445675213997171068, 4.21455659274653560642816607563, 4.39356677334090133886398563412, 4.45685587889652391463976540480, 4.99589375078119865612437326590, 5.01806034025202991336412962576, 5.10827318917029329979696675003, 5.26723511887452455028540333603, 5.62127766350659225704815069152, 5.78076389487534389143791738107, 5.94612734004827364700429546065, 6.07534980089151574561857576228, 6.52305652331659294763790874880

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