Properties

Label 8-637e4-1.1-c1e4-0-24
Degree $8$
Conductor $164648481361$
Sign $1$
Analytic cond. $669.369$
Root an. cond. $2.25532$
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  + 3·2-s + 5·4-s + 6·8-s + 9-s + 6·11-s + 4·13-s + 4·16-s − 6·17-s + 3·18-s + 6·19-s + 18·22-s + 12·23-s + 5·25-s + 12·26-s + 10·31-s − 18·34-s + 5·36-s + 4·37-s + 18·38-s − 32·43-s + 30·44-s + 36·46-s − 6·47-s + 15·50-s + 20·52-s + 6·53-s + 6·59-s + ⋯
L(s)  = 1  + 2.12·2-s + 5/2·4-s + 2.12·8-s + 1/3·9-s + 1.80·11-s + 1.10·13-s + 16-s − 1.45·17-s + 0.707·18-s + 1.37·19-s + 3.83·22-s + 2.50·23-s + 25-s + 2.35·26-s + 1.79·31-s − 3.08·34-s + 5/6·36-s + 0.657·37-s + 2.91·38-s − 4.87·43-s + 4.52·44-s + 5.30·46-s − 0.875·47-s + 2.12·50-s + 2.77·52-s + 0.824·53-s + 0.781·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(7^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(669.369\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{637} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 7^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(15.69775001\)
\(L(\frac12)\) \(\approx\) \(15.69775001\)
\(L(\frac{3}{2})\) 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)$
bad7 \( 1 \)
13$C_1$ \( ( 1 - T )^{4} \)
good2$C_2^2$$\times$$C_2^2$ \( ( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} )( 1 - T^{2} + p^{2} T^{4} ) \)
3$C_2^3$ \( 1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} \)
5$C_2$$\times$$C_2^2$ \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \)
11$C_2^2$ \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 6 T + 13 T^{2} - 66 T^{3} - 372 T^{4} - 66 p T^{5} + 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 12 T + 67 T^{2} - 372 T^{3} + 2088 T^{4} - 372 p T^{5} + 67 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 4 T - 17 T^{2} + 164 T^{3} - 872 T^{4} + 164 p T^{5} - 17 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
47$D_4\times C_2$ \( 1 + 6 T - p T^{2} - 66 T^{3} + 2988 T^{4} - 66 p T^{5} - p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 6 T - 59 T^{2} + 66 T^{3} + 4308 T^{4} + 66 p T^{5} - 59 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 6 T - 71 T^{2} + 66 T^{3} + 5844 T^{4} + 66 p T^{5} - 71 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 8 T - 53 T^{2} + 232 T^{3} + 3688 T^{4} + 232 p T^{5} - 53 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 8 T - 65 T^{2} - 232 T^{3} + 5344 T^{4} - 232 p T^{5} - 65 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2^3$ \( 1 - 173 T^{2} + 22008 T^{4} - 173 p^{2} T^{6} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.29659759714342890931595266705, −7.07142712545683308407459715000, −6.93076587463328469304648565271, −6.91975938625982184569873604364, −6.63557916723416195666476517112, −6.43656087010451697196656727330, −6.11586339675428518223589975536, −6.06978223505032457584478250949, −5.71177127466275601571271082886, −5.09806230723001197364171513873, −5.07530185526766253034894590276, −4.90336449348399530239699003410, −4.84654172195487196766523018450, −4.45700126473678639780881347558, −4.07349417103731306166888072790, −3.95607249836668106297901538249, −3.58190473073688495419791505823, −3.22908109928343167093342924606, −3.20905148485737465340353365424, −2.91831266741466623778743797806, −2.45378857554826320941896047546, −1.97072174160825553029468564769, −1.65101066267164557495071182343, −1.07959789309069802873356311898, −0.949291099823486747223309816262, 0.949291099823486747223309816262, 1.07959789309069802873356311898, 1.65101066267164557495071182343, 1.97072174160825553029468564769, 2.45378857554826320941896047546, 2.91831266741466623778743797806, 3.20905148485737465340353365424, 3.22908109928343167093342924606, 3.58190473073688495419791505823, 3.95607249836668106297901538249, 4.07349417103731306166888072790, 4.45700126473678639780881347558, 4.84654172195487196766523018450, 4.90336449348399530239699003410, 5.07530185526766253034894590276, 5.09806230723001197364171513873, 5.71177127466275601571271082886, 6.06978223505032457584478250949, 6.11586339675428518223589975536, 6.43656087010451697196656727330, 6.63557916723416195666476517112, 6.91975938625982184569873604364, 6.93076587463328469304648565271, 7.07142712545683308407459715000, 7.29659759714342890931595266705

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