Properties

Label 8-637e4-1.1-c1e4-0-27
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·4-s − 3·5-s + 9-s + 9·11-s − 3·12-s + 14·13-s + 3·15-s + 4·16-s + 12·17-s + 9·19-s − 9·20-s + 12·23-s − 3·25-s + 4·27-s + 9·29-s − 30·31-s − 9·33-s + 3·36-s − 14·39-s + 18·41-s − 5·43-s + 27·44-s − 3·45-s − 24·47-s − 4·48-s − 12·51-s + ⋯
L(s)  = 1  − 0.577·3-s + 3/2·4-s − 1.34·5-s + 1/3·9-s + 2.71·11-s − 0.866·12-s + 3.88·13-s + 0.774·15-s + 16-s + 2.91·17-s + 2.06·19-s − 2.01·20-s + 2.50·23-s − 3/5·25-s + 0.769·27-s + 1.67·29-s − 5.38·31-s − 1.56·33-s + 1/2·36-s − 2.24·39-s + 2.81·41-s − 0.762·43-s + 4.07·44-s − 0.447·45-s − 3.50·47-s − 0.577·48-s − 1.68·51-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

\(L(1)\) \(\approx\) \(6.323170502\)
\(L(\frac12)\) \(\approx\) \(6.323170502\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
good2$C_2^2$$\times$$C_2^2$ \( ( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} ) \)
3$D_4\times C_2$ \( 1 + T - 5 T^{3} - 11 T^{4} - 5 p T^{5} + p^{3} T^{7} + p^{4} T^{8} \)
5$D_4\times C_2$ \( 1 + 3 T + 12 T^{2} + 27 T^{3} + 71 T^{4} + 27 p T^{5} + 12 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 9 T + 54 T^{2} - 243 T^{3} + 905 T^{4} - 243 p T^{5} + 54 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{4} \)
19$D_4\times C_2$ \( 1 - 9 T + 56 T^{2} - 261 T^{3} + 993 T^{4} - 261 p T^{5} + 56 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 - 9 T + 8 T^{2} - 135 T^{3} + 2139 T^{4} - 135 p T^{5} + 8 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
41$D_4\times C_2$ \( 1 - 18 T + 210 T^{2} - 1836 T^{3} + 13151 T^{4} - 1836 p T^{5} + 210 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 5 T - 20 T^{2} - 205 T^{3} - 899 T^{4} - 205 p T^{5} - 20 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 + 24 T + 327 T^{2} + 3240 T^{3} + 25040 T^{4} + 3240 p T^{5} + 327 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 6 T + 5 T^{2} + 450 T^{3} - 3756 T^{4} + 450 p T^{5} + 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 6 T^{2} + 5627 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 2 T + 70 T^{2} + 376 T^{3} + 391 T^{4} + 376 p T^{5} + 70 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + 4 T + p T^{2} )^{2}( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
71$D_4\times C_2$ \( 1 - 6 T + 150 T^{2} - 828 T^{3} + 14855 T^{4} - 828 p T^{5} + 150 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 270 T^{2} + 31667 T^{4} - 270 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 87 T^{2} + 1853 T^{4} - 87 p^{2} T^{6} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 39 T + 812 T^{2} + 11895 T^{3} + 132795 T^{4} + 11895 p T^{5} + 812 p^{2} T^{6} + 39 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.60432479327146753986803404896, −7.31145815459036313551312016574, −7.04585765583933580504961077853, −6.82152337185676039283662164618, −6.70849290467544021938648239693, −6.64517179320774365373490598418, −6.23294361841214697552853423746, −6.02754290170039811582021070218, −5.58091600014260064667609010277, −5.49956221424824451192111860178, −5.42908137548725399376775896928, −5.23009486497388365542472260837, −4.56578541770108484399372854762, −4.10882791166648357287218098820, −4.06722926122427706905716311439, −3.65128888613961709687098563126, −3.56650979106340368876028914644, −3.33473671278930713498370274384, −3.19851392988863822957567174270, −2.97442661593573578698350622305, −2.08721821007536886643836102810, −1.56922570399140501144258352674, −1.22207996422282042602728488688, −1.14960984070049545214224612595, −1.06001833414365627601565033938, 1.06001833414365627601565033938, 1.14960984070049545214224612595, 1.22207996422282042602728488688, 1.56922570399140501144258352674, 2.08721821007536886643836102810, 2.97442661593573578698350622305, 3.19851392988863822957567174270, 3.33473671278930713498370274384, 3.56650979106340368876028914644, 3.65128888613961709687098563126, 4.06722926122427706905716311439, 4.10882791166648357287218098820, 4.56578541770108484399372854762, 5.23009486497388365542472260837, 5.42908137548725399376775896928, 5.49956221424824451192111860178, 5.58091600014260064667609010277, 6.02754290170039811582021070218, 6.23294361841214697552853423746, 6.64517179320774365373490598418, 6.70849290467544021938648239693, 6.82152337185676039283662164618, 7.04585765583933580504961077853, 7.31145815459036313551312016574, 7.60432479327146753986803404896

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