Properties

Degree $8$
Conductor $1.626\times 10^{13}$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 2·3-s + 10·4-s − 8·6-s + 20·8-s + 9-s + 3·11-s − 20·12-s + 35·16-s − 2·17-s + 4·18-s − 2·19-s + 12·22-s − 40·24-s + 4·25-s + 56·32-s − 6·33-s − 8·34-s + 10·36-s − 8·38-s − 2·41-s − 2·43-s + 30·44-s − 70·48-s − 49-s + 16·50-s + 4·51-s + ⋯
L(s)  = 1  + 4·2-s − 2·3-s + 10·4-s − 8·6-s + 20·8-s + 9-s + 3·11-s − 20·12-s + 35·16-s − 2·17-s + 4·18-s − 2·19-s + 12·22-s − 40·24-s + 4·25-s + 56·32-s − 6·33-s − 8·34-s + 10·36-s − 8·38-s − 2·41-s − 2·43-s + 30·44-s − 70·48-s − 49-s + 16·50-s + 4·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 251^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 251^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 251^{4}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{2008} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 251^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(13.37437553\)
\(L(\frac12)\) \(\approx\) \(13.37437553\)
\(L(1)\) 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$C_1$ \( ( 1 - T )^{4} \)
251$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
good3$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
7$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
11$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
17$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
19$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
29$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
31$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
41$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
53$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
59$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
61$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
67$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
71$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
73$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
79$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
89$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
97$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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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

−6.66366075442673146063393648845, −6.27357892238035025396747281397, −6.17586723183085153684699797146, −6.13776501237703708797100137191, −5.98320338710540756900179282016, −5.55442959367847410153223555095, −5.28747670265424239803874419285, −5.20229993260623766015317565465, −5.08780076158096774088830567224, −4.59661846145463986290395199498, −4.58402567721183794783402779099, −4.53255540667671884101917998738, −4.43798917687498245782455892063, −3.82629587850914389270770732306, −3.74758887455869523144583677496, −3.66792740311743405604467278088, −3.43639917452265306101856813983, −2.96872576047878490064981224139, −2.82380847350095772688315132651, −2.47263942200768363886577483638, −2.28945656651594676027275666223, −1.89151238285185649856963161542, −1.53126557955658848634939012762, −1.36425799060875297604548973382, −1.10018684094431658516196510905, 1.10018684094431658516196510905, 1.36425799060875297604548973382, 1.53126557955658848634939012762, 1.89151238285185649856963161542, 2.28945656651594676027275666223, 2.47263942200768363886577483638, 2.82380847350095772688315132651, 2.96872576047878490064981224139, 3.43639917452265306101856813983, 3.66792740311743405604467278088, 3.74758887455869523144583677496, 3.82629587850914389270770732306, 4.43798917687498245782455892063, 4.53255540667671884101917998738, 4.58402567721183794783402779099, 4.59661846145463986290395199498, 5.08780076158096774088830567224, 5.20229993260623766015317565465, 5.28747670265424239803874419285, 5.55442959367847410153223555095, 5.98320338710540756900179282016, 6.13776501237703708797100137191, 6.17586723183085153684699797146, 6.27357892238035025396747281397, 6.66366075442673146063393648845

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