Properties

Degree $8$
Conductor $2.479\times 10^{15}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·19-s − 10·25-s − 4·31-s + 16·37-s + 32·103-s − 16·109-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 3.67·19-s − 2·25-s − 0.718·31-s + 2.63·37-s + 3.15·103-s − 1.53·109-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{7056} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.975605915\)
\(L(\frac12)\) \(\approx\) \(3.975605915\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 - p T^{2} )^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 13 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 61 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 73 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 121 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )^{2}( 1 + 19 T + p T^{2} )^{2} \)
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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

−5.60357944718570714092716278009, −5.51979078214773733092702489143, −5.42027437219794824652123258076, −4.85201633656257562332589149150, −4.80644378787604916477876619009, −4.68115378228741512986835231101, −4.61173007060716447557845031909, −4.15482889882680533426857102683, −4.10755386262384893302972959422, −3.82699095089392743476653733585, −3.68812297127066742060218331275, −3.26149578250211477175875619857, −3.23744140325021410495133196844, −3.20988357544793288953886458600, −3.03865359479527171503023370452, −2.43026732656221220935317623142, −2.35646533275950541110443070020, −2.30549641026596195091709438638, −2.02212308629211435461448464649, −1.53504394174330797523206353312, −1.42725942948287277683751612130, −1.13749648477910573252393607553, −0.908493481907067321896044956712, −0.63329011669784976241150922127, −0.24753622179303892227757987686, 0.24753622179303892227757987686, 0.63329011669784976241150922127, 0.908493481907067321896044956712, 1.13749648477910573252393607553, 1.42725942948287277683751612130, 1.53504394174330797523206353312, 2.02212308629211435461448464649, 2.30549641026596195091709438638, 2.35646533275950541110443070020, 2.43026732656221220935317623142, 3.03865359479527171503023370452, 3.20988357544793288953886458600, 3.23744140325021410495133196844, 3.26149578250211477175875619857, 3.68812297127066742060218331275, 3.82699095089392743476653733585, 4.10755386262384893302972959422, 4.15482889882680533426857102683, 4.61173007060716447557845031909, 4.68115378228741512986835231101, 4.80644378787604916477876619009, 4.85201633656257562332589149150, 5.42027437219794824652123258076, 5.51979078214773733092702489143, 5.60357944718570714092716278009

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