Properties

Label 8-648e4-1.1-c3e4-0-3
Degree $8$
Conductor $176319369216$
Sign $1$
Analytic cond. $2.13680\times 10^{6}$
Root an. cond. $6.18330$
Motivic weight $3$
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  + 4·5-s + 6·7-s + 52·11-s − 26·13-s − 376·17-s − 148·19-s + 148·23-s + 74·25-s + 288·29-s + 248·31-s + 24·35-s + 684·37-s + 256·43-s + 132·47-s − 25·49-s − 1.90e3·53-s + 208·55-s − 1.00e3·59-s + 34·61-s − 104·65-s + 866·67-s − 1.55e3·71-s + 3.74e3·73-s + 312·77-s − 182·79-s − 1.33e3·83-s − 1.50e3·85-s + ⋯
L(s)  = 1  + 0.357·5-s + 0.323·7-s + 1.42·11-s − 0.554·13-s − 5.36·17-s − 1.78·19-s + 1.34·23-s + 0.591·25-s + 1.84·29-s + 1.43·31-s + 0.115·35-s + 3.03·37-s + 0.907·43-s + 0.409·47-s − 0.0728·49-s − 4.93·53-s + 0.509·55-s − 2.21·59-s + 0.0713·61-s − 0.198·65-s + 1.57·67-s − 2.59·71-s + 6.00·73-s + 0.461·77-s − 0.259·79-s − 1.76·83-s − 1.91·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(2.13680\times 10^{6}\)
Root analytic conductor: \(6.18330\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{648} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{16} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(1.841486597\)
\(L(\frac12)\) \(\approx\) \(1.841486597\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5$D_4\times C_2$ \( 1 - 4 T - 58 T^{2} + 704 T^{3} - 12149 T^{4} + 704 p^{3} T^{5} - 58 p^{6} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \)
7$D_4\times C_2$ \( 1 - 6 T + 61 T^{2} + 4266 T^{3} - 129372 T^{4} + 4266 p^{3} T^{5} + 61 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 - 52 T + 986 T^{2} + 49088 T^{3} - 2419061 T^{4} + 49088 p^{3} T^{5} + 986 p^{6} T^{6} - 52 p^{9} T^{7} + p^{12} T^{8} \)
13$D_4\times C_2$ \( 1 + 2 p T - 3167 T^{2} - 1102 p T^{3} + 8456668 T^{4} - 1102 p^{4} T^{5} - 3167 p^{6} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \)
17$D_{4}$ \( ( 1 + 188 T + 18482 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 + 74 T + 3567 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 148 T - 3406 T^{2} - 144448 T^{3} + 226054243 T^{4} - 144448 p^{3} T^{5} - 3406 p^{6} T^{6} - 148 p^{9} T^{7} + p^{12} T^{8} \)
29$D_4\times C_2$ \( 1 - 288 T + 24950 T^{2} - 2654208 T^{3} + 745559499 T^{4} - 2654208 p^{3} T^{5} + 24950 p^{6} T^{6} - 288 p^{9} T^{7} + p^{12} T^{8} \)
31$D_4\times C_2$ \( 1 - 8 p T - 1934 T^{2} - 30848 p T^{3} + 1304610499 T^{4} - 30848 p^{4} T^{5} - 1934 p^{6} T^{6} - 8 p^{10} T^{7} + p^{12} T^{8} \)
37$D_{4}$ \( ( 1 - 342 T + 112547 T^{2} - 342 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
41$C_2^3$ \( 1 + 46478 T^{2} - 2589899757 T^{4} + 46478 p^{6} T^{6} + p^{12} T^{8} \)
43$D_4\times C_2$ \( 1 - 256 T - 106982 T^{2} - 3457024 T^{3} + 18230526523 T^{4} - 3457024 p^{3} T^{5} - 106982 p^{6} T^{6} - 256 p^{9} T^{7} + p^{12} T^{8} \)
47$D_4\times C_2$ \( 1 - 132 T - 194398 T^{2} - 551232 T^{3} + 32280332403 T^{4} - 551232 p^{3} T^{5} - 194398 p^{6} T^{6} - 132 p^{9} T^{7} + p^{12} T^{8} \)
53$D_{4}$ \( ( 1 + 952 T + 489050 T^{2} + 952 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 1004 T + 424634 T^{2} + 173314496 T^{3} + 91128706219 T^{4} + 173314496 p^{3} T^{5} + 424634 p^{6} T^{6} + 1004 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 - 34 T - 245015 T^{2} + 7064894 T^{3} + 8817396844 T^{4} + 7064894 p^{3} T^{5} - 245015 p^{6} T^{6} - 34 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 - 866 T + 7021 T^{2} - 122460194 T^{3} + 235935015628 T^{4} - 122460194 p^{3} T^{5} + 7021 p^{6} T^{6} - 866 p^{9} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 + 776 T + 704366 T^{2} + 776 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 - 1874 T + 1630083 T^{2} - 1874 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 + 182 T - 839555 T^{2} - 20638618 T^{3} + 502149757684 T^{4} - 20638618 p^{3} T^{5} - 839555 p^{6} T^{6} + 182 p^{9} T^{7} + p^{12} T^{8} \)
83$D_4\times C_2$ \( 1 + 1336 T + 575978 T^{2} + 87299584 T^{3} + 113962028203 T^{4} + 87299584 p^{3} T^{5} + 575978 p^{6} T^{6} + 1336 p^{9} T^{7} + p^{12} T^{8} \)
89$D_{4}$ \( ( 1 + 876 T + 1522402 T^{2} + 876 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 38 T - 430343 T^{2} + 52955242 T^{3} - 647849891372 T^{4} + 52955242 p^{3} T^{5} - 430343 p^{6} T^{6} - 38 p^{9} T^{7} + p^{12} T^{8} \)
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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

−7.04785972559845358109066781381, −6.75233890575325223294881566161, −6.62092583504775877407188485696, −6.48306748116878344935155062519, −6.22567769541823806087956936595, −6.21079044345563968752056599504, −6.19979711591800297982765225212, −5.38603532394268862074831551448, −4.88933674352806933577183375269, −4.88734942728512242051211416755, −4.85410802747072247727010913998, −4.27886286454434475181516457246, −4.26895116756944567091658777333, −4.15942588834132507394185912703, −4.01442633695128299123600373140, −3.19332706348158947710234427559, −2.79386925710880524474666531158, −2.63776854324253475319595232904, −2.59969729938978975763646384272, −2.15910359504680760512463596943, −1.84436103792039187991086322047, −1.32494714686276510785778733033, −1.23440485963593287961864539033, −0.51913460457170857065425030902, −0.22651305048014991350625963652, 0.22651305048014991350625963652, 0.51913460457170857065425030902, 1.23440485963593287961864539033, 1.32494714686276510785778733033, 1.84436103792039187991086322047, 2.15910359504680760512463596943, 2.59969729938978975763646384272, 2.63776854324253475319595232904, 2.79386925710880524474666531158, 3.19332706348158947710234427559, 4.01442633695128299123600373140, 4.15942588834132507394185912703, 4.26895116756944567091658777333, 4.27886286454434475181516457246, 4.85410802747072247727010913998, 4.88734942728512242051211416755, 4.88933674352806933577183375269, 5.38603532394268862074831551448, 6.19979711591800297982765225212, 6.21079044345563968752056599504, 6.22567769541823806087956936595, 6.48306748116878344935155062519, 6.62092583504775877407188485696, 6.75233890575325223294881566161, 7.04785972559845358109066781381

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