Properties

Label 8-154e4-1.1-c1e4-0-1
Degree $8$
Conductor $562448656$
Sign $1$
Analytic cond. $2.28660$
Root an. cond. $1.10891$
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  + 2·2-s + 4-s − 2·5-s − 2·8-s − 9-s − 4·10-s − 2·11-s + 20·13-s − 4·16-s − 12·17-s − 2·18-s + 6·19-s − 2·20-s − 4·22-s + 2·23-s + 4·25-s + 40·26-s + 4·29-s + 8·31-s − 2·32-s − 24·34-s − 36-s − 2·37-s + 12·38-s + 4·40-s + 12·41-s − 16·43-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s − 0.894·5-s − 0.707·8-s − 1/3·9-s − 1.26·10-s − 0.603·11-s + 5.54·13-s − 16-s − 2.91·17-s − 0.471·18-s + 1.37·19-s − 0.447·20-s − 0.852·22-s + 0.417·23-s + 4/5·25-s + 7.84·26-s + 0.742·29-s + 1.43·31-s − 0.353·32-s − 4.11·34-s − 1/6·36-s − 0.328·37-s + 1.94·38-s + 0.632·40-s + 1.87·41-s − 2.43·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(2.28660\)
Root analytic conductor: \(1.10891\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{154} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.297646822\)
\(L(\frac12)\) \(\approx\) \(2.297646822\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( ( 1 - T + T^{2} )^{2} \)
7$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + T + T^{2} )^{2} \)
good3$C_2^3$ \( 1 + T^{2} - 8 T^{4} + p^{2} T^{6} + p^{4} T^{8} \)
5$D_4\times C_2$ \( 1 + 2 T - 12 T^{3} - 29 T^{4} - 12 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 6 T - 4 T^{2} - 12 T^{3} + 555 T^{4} - 12 p T^{5} - 4 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 2 T - 36 T^{2} + 12 T^{3} + 979 T^{4} + 12 p T^{5} - 36 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 2 T + 31 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
37$D_4\times C_2$ \( 1 + 2 T - 64 T^{2} - 12 T^{3} + 3107 T^{4} - 12 p T^{5} - 64 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
47$D_4\times C_2$ \( 1 + 16 T + 126 T^{2} + 576 T^{3} + 2659 T^{4} + 576 p T^{5} + 126 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 2 T - 96 T^{2} + 12 T^{3} + 6979 T^{4} + 12 p T^{5} - 96 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 4 T - 99 T^{2} - 12 T^{3} + 8800 T^{4} - 12 p T^{5} - 99 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 18 T + 149 T^{2} + 954 T^{3} + 7140 T^{4} + 954 p T^{5} + 149 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 8 T - 23 T^{2} + 376 T^{3} - 1208 T^{4} + 376 p T^{5} - 23 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 14 T + 184 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 6 T - 112 T^{2} + 12 T^{3} + 13947 T^{4} + 12 p T^{5} - 112 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^3$ \( 1 - 151 T^{2} + 16560 T^{4} - 151 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 8 T - 18 T^{2} + 768 T^{3} - 6893 T^{4} + 768 p T^{5} - 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 + 22 T + 287 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−9.430245649503273250634322470650, −9.169116142552141687309255412640, −9.034082311170186691341034692542, −8.550679611979928545023305608530, −8.346903938502269848788683500109, −8.134117974988046124052436441104, −8.021986884520513601419562775696, −7.968035872748558359004354378445, −6.88983008761910233471025359542, −6.69797574880814778690093801995, −6.53744348665975916969101268034, −6.50950775462488720817289925770, −5.96478698770643686587908090181, −5.93403474009816346402816364224, −5.26612020791751810280609560819, −5.02323960448957484868037720839, −4.70840771771047264601916013173, −4.41205116391757646211617602553, −3.79638292825034123679365534386, −3.73725266090806095837061049493, −3.57450956170679545052740211294, −2.96298327526848233382819988489, −2.77152966487750449609004297313, −1.68186501093571855182183849520, −1.08928941804182991840182327617, 1.08928941804182991840182327617, 1.68186501093571855182183849520, 2.77152966487750449609004297313, 2.96298327526848233382819988489, 3.57450956170679545052740211294, 3.73725266090806095837061049493, 3.79638292825034123679365534386, 4.41205116391757646211617602553, 4.70840771771047264601916013173, 5.02323960448957484868037720839, 5.26612020791751810280609560819, 5.93403474009816346402816364224, 5.96478698770643686587908090181, 6.50950775462488720817289925770, 6.53744348665975916969101268034, 6.69797574880814778690093801995, 6.88983008761910233471025359542, 7.968035872748558359004354378445, 8.021986884520513601419562775696, 8.134117974988046124052436441104, 8.346903938502269848788683500109, 8.550679611979928545023305608530, 9.034082311170186691341034692542, 9.169116142552141687309255412640, 9.430245649503273250634322470650

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