# 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 $4$

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## 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