Properties

Label 8-96e8-1.1-c1e4-0-13
Degree $8$
Conductor $7.214\times 10^{15}$
Sign $1$
Analytic cond. $2.93277\times 10^{7}$
Root an. cond. $8.57846$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·5-s + 8·13-s + 24·25-s − 24·29-s + 8·37-s + 16·41-s − 12·49-s − 24·53-s + 24·61-s − 64·65-s + 8·73-s − 8·89-s + 32·97-s − 40·101-s − 24·109-s + 8·113-s − 4·121-s − 8·125-s + 127-s + 131-s + 137-s + 139-s + 192·145-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 3.57·5-s + 2.21·13-s + 24/5·25-s − 4.45·29-s + 1.31·37-s + 2.49·41-s − 1.71·49-s − 3.29·53-s + 3.07·61-s − 7.93·65-s + 0.936·73-s − 0.847·89-s + 3.24·97-s − 3.98·101-s − 2.29·109-s + 0.752·113-s − 0.363·121-s − 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 15.9·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{40} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(2.93277\times 10^{7}\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{40} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$D_{4}$ \( ( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
7$C_2^2:C_4$ \( 1 + 12 T^{2} + 102 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2:C_4$ \( 1 + 4 T^{2} + 238 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
13$D_{4}$ \( ( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2:C_4$ \( 1 + 36 T^{2} + 654 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2:C_4$ \( 1 + 12 T^{2} + 1062 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 12 T + 92 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2:C_4$ \( 1 + 60 T^{2} + 2310 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
43$C_2^2:C_4$ \( 1 + 164 T^{2} + 10414 T^{4} + 164 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2:C_4$ \( 1 + 124 T^{2} + 7750 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 + 12 T + 140 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2:C_4$ \( 1 + 228 T^{2} + 19950 T^{4} + 228 p^{2} T^{6} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 - 12 T + 108 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2:C_4$ \( 1 + 164 T^{2} + 13390 T^{4} + 164 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2:C_4$ \( 1 + 76 T^{2} + 9958 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \)
73$D_{4}$ \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2:C_4$ \( 1 + 60 T^{2} + 5190 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2^2:C_4$ \( 1 + 132 T^{2} + 10446 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 4 T + 174 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 16 T + 250 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−5.90165196871777992755349370870, −5.58682606603423456858496520709, −5.30831334700932836176600640147, −5.24584881398966307612519510281, −5.12800502804161011687748320924, −4.68123625042312939000019437908, −4.47805994756070011413855338964, −4.37566759450764295616120560281, −4.27190358194255586291093686200, −4.10764818238172151577803702751, −3.79646757050646788144536410405, −3.77193287918665618326138021370, −3.60076901907742153906142675220, −3.46329845077836699306993877127, −3.28639789077989039084973785012, −3.26885827845665234564409895407, −2.92469802377438730878936135794, −2.40830292287460573938227706305, −2.19232705599899405750085131366, −2.15789399517423854871645815589, −2.05539576898860840693665372343, −1.32222178361332820850877716905, −1.18085164726689439997569162348, −1.15797141791314107038197574115, −0.925189404623596474063353430682, 0, 0, 0, 0, 0.925189404623596474063353430682, 1.15797141791314107038197574115, 1.18085164726689439997569162348, 1.32222178361332820850877716905, 2.05539576898860840693665372343, 2.15789399517423854871645815589, 2.19232705599899405750085131366, 2.40830292287460573938227706305, 2.92469802377438730878936135794, 3.26885827845665234564409895407, 3.28639789077989039084973785012, 3.46329845077836699306993877127, 3.60076901907742153906142675220, 3.77193287918665618326138021370, 3.79646757050646788144536410405, 4.10764818238172151577803702751, 4.27190358194255586291093686200, 4.37566759450764295616120560281, 4.47805994756070011413855338964, 4.68123625042312939000019437908, 5.12800502804161011687748320924, 5.24584881398966307612519510281, 5.30831334700932836176600640147, 5.58682606603423456858496520709, 5.90165196871777992755349370870

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