Properties

Label 8-1792e4-1.1-c1e4-0-16
Degree $8$
Conductor $1.031\times 10^{13}$
Sign $1$
Analytic cond. $41923.7$
Root an. cond. $3.78274$
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  + 4·3-s + 8·7-s + 4·9-s + 12·19-s + 32·21-s + 12·25-s − 4·27-s − 16·37-s + 34·49-s + 48·57-s + 36·59-s + 32·63-s + 48·75-s − 10·81-s + 36·83-s + 48·103-s − 64·111-s − 24·113-s + 12·121-s + 127-s + 131-s + 96·133-s + 137-s + 139-s + 136·147-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2.30·3-s + 3.02·7-s + 4/3·9-s + 2.75·19-s + 6.98·21-s + 12/5·25-s − 0.769·27-s − 2.63·37-s + 34/7·49-s + 6.35·57-s + 4.68·59-s + 4.03·63-s + 5.54·75-s − 1.11·81-s + 3.95·83-s + 4.72·103-s − 6.07·111-s − 2.25·113-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 8.32·133-s + 0.0854·137-s + 0.0848·139-s + 11.2·147-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(41923.7\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1792} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(24.77964951\)
\(L(\frac12)\) \(\approx\) \(24.77964951\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
good3$D_{4}$ \( ( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
5$D_4\times C_2$ \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 - 28 T^{2} + 426 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
41$D_4\times C_2$ \( 1 - 36 T^{2} + 614 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 76 T^{2} + 3414 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
59$D_{4}$ \( ( 1 - 18 T + 172 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - 76 T^{2} + 6186 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 172 T^{2} + 14646 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 132 T^{2} + 11366 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \)
83$D_{4}$ \( ( 1 - 18 T + 244 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 132 T^{2} + 7910 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.65127397876537475189678540714, −6.54828033878958475927840619234, −6.39817283727256316719675304594, −5.68269782999461804617552959280, −5.57940187706500157296648458374, −5.36503800484731154417755773162, −5.25055608999607256001843169193, −5.19964954310748762484086287959, −4.94813385345460826016104361156, −4.62132978223779372232419129059, −4.47728641549432196620877133598, −4.18918020746038407995323999605, −3.83583068349452777716266675834, −3.56236640555768573406710884944, −3.24895834688436860532750468647, −3.22002225352805589147018759302, −3.15937721835126699757389096032, −2.64157986983474498769625291627, −2.37961169605677306685158279935, −2.12590284817796909370800595803, −1.93689706775363467564271433827, −1.79939323242850250828922193375, −1.07096604102383221240591265392, −0.996521312305338989560089000490, −0.78831150340855633856001371166, 0.78831150340855633856001371166, 0.996521312305338989560089000490, 1.07096604102383221240591265392, 1.79939323242850250828922193375, 1.93689706775363467564271433827, 2.12590284817796909370800595803, 2.37961169605677306685158279935, 2.64157986983474498769625291627, 3.15937721835126699757389096032, 3.22002225352805589147018759302, 3.24895834688436860532750468647, 3.56236640555768573406710884944, 3.83583068349452777716266675834, 4.18918020746038407995323999605, 4.47728641549432196620877133598, 4.62132978223779372232419129059, 4.94813385345460826016104361156, 5.19964954310748762484086287959, 5.25055608999607256001843169193, 5.36503800484731154417755773162, 5.57940187706500157296648458374, 5.68269782999461804617552959280, 6.39817283727256316719675304594, 6.54828033878958475927840619234, 6.65127397876537475189678540714

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