Properties

Label 8-2352e4-1.1-c1e4-0-1
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $124410.$
Root an. cond. $4.33368$
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 + 10·9-s + 16·25-s − 20·27-s − 16·29-s + 16·31-s − 16·47-s − 16·53-s − 64·75-s + 35·81-s − 48·83-s + 64·87-s − 64·93-s − 48·103-s + 32·109-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 64·141-s + 149-s + 151-s + 157-s + 64·159-s + 163-s + 167-s + ⋯
L(s)  = 1  − 2.30·3-s + 10/3·9-s + 16/5·25-s − 3.84·27-s − 2.97·29-s + 2.87·31-s − 2.33·47-s − 2.19·53-s − 7.39·75-s + 35/9·81-s − 5.26·83-s + 6.86·87-s − 6.63·93-s − 4.72·103-s + 3.06·109-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.38·141-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 5.07·159-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(124410.\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1045989007\)
\(L(\frac12)\) \(\approx\) \(0.1045989007\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + T )^{4} \)
7 \( 1 \)
good5$C_2^2:C_4$ \( 1 - 16 T^{2} + 112 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2:C_4$ \( 1 - 12 T^{2} + 150 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2:C_4$ \( 1 - 32 T^{2} + 496 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \)
17$C_4\times C_2$ \( 1 - 48 T^{2} + 1152 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2:C_4$ \( 1 - 60 T^{2} + 1830 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 72 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2:C_4$ \( 1 - 112 T^{2} + 6400 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2^2:C_4$ \( 1 - 108 T^{2} + 6102 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2:C_4$ \( 1 - 192 T^{2} + 16080 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2:C_4$ \( 1 + 52 T^{2} + 9142 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2:C_4$ \( 1 + 4 T^{2} - 282 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^2:C_4$ \( 1 - 224 T^{2} + 23104 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2:C_4$ \( 1 + 4 T^{2} + 11974 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \)
89$C_2^2:C_4$ \( 1 - 144 T^{2} + 20448 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2:C_4$ \( 1 - 288 T^{2} + 38304 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.32647727843282047794853010092, −6.30841562512505362466084337100, −6.03538927915580055116666514504, −5.71520306341466831760459369912, −5.52463584195135138291247305371, −5.32343922505680147297635481476, −5.20873921394203099062030772232, −4.87967084877586384608896834488, −4.76040216370376423020191417908, −4.57977351298623338823676199041, −4.52431060928792218274667896972, −4.08237623411166074745723648960, −3.86233774080046789716250612311, −3.74671946656088102014515150209, −3.29489312991416412464782763490, −3.08725971063358345708231687965, −2.82379802445324777454984750692, −2.52464494927919617870615438511, −2.48449953552213712499924520481, −1.61049140143573995827314163094, −1.56566591874379821537916830576, −1.42291480824776091817026402485, −1.11456933102686322758731587252, −0.60334433333222168708839991967, −0.087464364595941041859177282238, 0.087464364595941041859177282238, 0.60334433333222168708839991967, 1.11456933102686322758731587252, 1.42291480824776091817026402485, 1.56566591874379821537916830576, 1.61049140143573995827314163094, 2.48449953552213712499924520481, 2.52464494927919617870615438511, 2.82379802445324777454984750692, 3.08725971063358345708231687965, 3.29489312991416412464782763490, 3.74671946656088102014515150209, 3.86233774080046789716250612311, 4.08237623411166074745723648960, 4.52431060928792218274667896972, 4.57977351298623338823676199041, 4.76040216370376423020191417908, 4.87967084877586384608896834488, 5.20873921394203099062030772232, 5.32343922505680147297635481476, 5.52463584195135138291247305371, 5.71520306341466831760459369912, 6.03538927915580055116666514504, 6.30841562512505362466084337100, 6.32647727843282047794853010092

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