Properties

Degree $8$
Conductor $1.100\times 10^{12}$
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  + 8·17-s − 32·47-s − 4·49-s − 32·79-s + 14·81-s − 8·97-s − 8·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  + 1.94·17-s − 4.66·47-s − 4/7·49-s − 3.60·79-s + 14/9·81-s − 0.812·97-s − 0.752·113-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 + 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 + 0.0660·229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{40}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1024} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{40} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.8901810629\)
\(L(\frac12)\) \(\approx\) \(0.8901810629\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2$$\times$$C_2^2$ \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
5$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )( 1 + 8 T^{2} + p^{2} T^{4} ) \)
7$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^3$ \( 1 + 82 T^{4} + p^{4} T^{8} \)
13$C_2^3$ \( 1 + 146 T^{4} + p^{4} T^{8} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
19$C_2^2$$\times$$C_2^2$ \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \)
23$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^3$ \( 1 - 1198 T^{4} + p^{4} T^{8} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2^3$ \( 1 - 2062 T^{4} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^3$ \( 1 - 1198 T^{4} + p^{4} T^{8} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
53$C_2^3$ \( 1 - 718 T^{4} + p^{4} T^{8} \)
59$C_2^3$ \( 1 - 878 T^{4} + p^{4} T^{8} \)
61$C_2^3$ \( 1 + 6482 T^{4} + p^{4} T^{8} \)
67$C_2^3$ \( 1 - 7822 T^{4} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
83$C_2^3$ \( 1 + 3122 T^{4} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 - 174 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
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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

−7.00880621959966076447059244682, −6.99761075894000849361771495442, −6.62333895088324518299342854291, −6.37714583895726910445856578997, −6.33061058516496070954588142833, −6.04929189231026863666023906150, −5.79554414405538721509947445420, −5.35633980321687366441404325791, −5.24456934365175119541046183114, −5.21100749556451135527477347663, −4.94674293624403602174610833580, −4.52070265912957728645985008856, −4.37316585694665343858710972664, −3.99698149468041112019614257119, −3.83788875370598494527992325940, −3.40427206199065111265939285842, −3.24030806216906782130917264120, −3.11869155992713791672063183052, −2.65285222605664471768876986082, −2.56983300073975949157135885651, −1.89091151206295870206328562803, −1.63751757047662800139132892455, −1.37037615342815881249183204910, −1.06757726028612270039533786595, −0.21513376187048329687215324420, 0.21513376187048329687215324420, 1.06757726028612270039533786595, 1.37037615342815881249183204910, 1.63751757047662800139132892455, 1.89091151206295870206328562803, 2.56983300073975949157135885651, 2.65285222605664471768876986082, 3.11869155992713791672063183052, 3.24030806216906782130917264120, 3.40427206199065111265939285842, 3.83788875370598494527992325940, 3.99698149468041112019614257119, 4.37316585694665343858710972664, 4.52070265912957728645985008856, 4.94674293624403602174610833580, 5.21100749556451135527477347663, 5.24456934365175119541046183114, 5.35633980321687366441404325791, 5.79554414405538721509947445420, 6.04929189231026863666023906150, 6.33061058516496070954588142833, 6.37714583895726910445856578997, 6.62333895088324518299342854291, 6.99761075894000849361771495442, 7.00880621959966076447059244682

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