Properties

Label 8-63e8-1.1-c1e4-0-3
Degree $8$
Conductor $2.482\times 10^{14}$
Sign $1$
Analytic cond. $1.00886\times 10^{6}$
Root an. cond. $5.62962$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 16·13-s + 4·16-s − 4·19-s + 12·37-s − 16·43-s + 48·52-s − 28·61-s − 9·64-s + 16·67-s − 24·73-s + 12·76-s − 16·79-s − 12·97-s + 4·103-s + 24·109-s − 30·121-s + 127-s + 131-s + 137-s + 139-s − 36·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 3/2·4-s − 4.43·13-s + 16-s − 0.917·19-s + 1.97·37-s − 2.43·43-s + 6.65·52-s − 3.58·61-s − 9/8·64-s + 1.95·67-s − 2.80·73-s + 1.37·76-s − 1.80·79-s − 1.21·97-s + 0.394·103-s + 2.29·109-s − 2.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.95·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.00886\times 10^{6}\)
Root analytic conductor: \(5.62962\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3969} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{16} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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 \( 1 \)
good2$C_2^3$ \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \)
5$C_2^3$ \( 1 - 34 T^{4} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + 15 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 + 36 T^{2} + 662 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{4} \)
41$D_4\times C_2$ \( 1 + 144 T^{2} + 8462 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 + 8 T + 81 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4$ \( 1 + 8 T^{2} - 2370 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 + 134 T^{2} + 11547 T^{4} + 134 p^{2} T^{6} + p^{4} T^{8} \)
73$D_{4}$ \( ( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 8 T + 153 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4$ \( 1 + 200 T^{2} + 23022 T^{4} + 200 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 + 276 T^{2} + 33542 T^{4} + 276 p^{2} T^{6} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 + 6 T + 182 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.27661711966345296585423927696, −6.20095176653184636785587238320, −5.94453364143308240053029029242, −5.67548358980191560063788235652, −5.55992144112937732567681453368, −5.16053292904869965214499364826, −5.10748005533649218667929718106, −4.94781420762872407099838493559, −4.78548770335863849134016757249, −4.61291145967534222187845065040, −4.42583864367306258948360895717, −4.30750808835169801451393281377, −4.16267519997863920585793762488, −3.75439530259414010244687602232, −3.62716218766247370534390205691, −3.23402317744906237772790240912, −2.94174418005259728102104280286, −2.87742802521184035723286903063, −2.65082430387052121284013116860, −2.30400285535455280236254022295, −2.27403512392638431120037072497, −1.94759042914898152857702265814, −1.52236847966008350898464181884, −1.18217712480094960309198168910, −1.05701004118786064889708925228, 0, 0, 0, 0, 1.05701004118786064889708925228, 1.18217712480094960309198168910, 1.52236847966008350898464181884, 1.94759042914898152857702265814, 2.27403512392638431120037072497, 2.30400285535455280236254022295, 2.65082430387052121284013116860, 2.87742802521184035723286903063, 2.94174418005259728102104280286, 3.23402317744906237772790240912, 3.62716218766247370534390205691, 3.75439530259414010244687602232, 4.16267519997863920585793762488, 4.30750808835169801451393281377, 4.42583864367306258948360895717, 4.61291145967534222187845065040, 4.78548770335863849134016757249, 4.94781420762872407099838493559, 5.10748005533649218667929718106, 5.16053292904869965214499364826, 5.55992144112937732567681453368, 5.67548358980191560063788235652, 5.94453364143308240053029029242, 6.20095176653184636785587238320, 6.27661711966345296585423927696

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