Properties

Label 8-96e8-1.1-c1e4-0-2
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·11-s + 8·17-s − 8·19-s − 8·41-s + 24·43-s − 4·49-s + 48·67-s − 24·73-s + 8·83-s + 40·89-s − 16·97-s − 32·107-s + 8·113-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 48·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2.41·11-s + 1.94·17-s − 1.83·19-s − 1.24·41-s + 3.65·43-s − 4/7·49-s + 5.86·67-s − 2.80·73-s + 0.878·83-s + 4.23·89-s − 1.62·97-s − 3.09·107-s + 0.752·113-s + 3.27·121-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 − 3.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{40} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.785348205\)
\(L(\frac12)\) \(\approx\) \(1.785348205\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 + 4 T^{2} + 22 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
11$C_4$ \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 + 80 T^{2} + 2962 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 + 4 T^{2} - 74 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 32 T^{2} + 114 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$D_4\times C_2$ \( 1 + 48 T^{2} + 3314 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 + 128 T^{2} + 8658 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
71$D_4\times C_2$ \( 1 + 188 T^{2} + 17638 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} \)
73$D_{4}$ \( ( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 + 260 T^{2} + 28662 T^{4} + 260 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
97$D_{4}$ \( ( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−5.43129413578133342196283885383, −5.34032789337258037949447130142, −4.96535600581706233654654978046, −4.93740213913496464232968464666, −4.89352609033869587113095039843, −4.48822459196255333159797730374, −4.21403388844508944825607083619, −4.15082971204870365831830797329, −3.96535071998907838090486600450, −3.64963271525450533347696601410, −3.58645152038127637363863219682, −3.45061999192101978908655795775, −3.09488433174261509832197533352, −2.77105187206083539127962618620, −2.61920292831551753897990457283, −2.61137784043725239779945589118, −2.51448665315435471446402228183, −1.94836447212783504927597628299, −1.93614479238635172144932758884, −1.80284805721405326829505587038, −1.38446150759953836530121938653, −0.927190181250870571167176206248, −0.72368989513177506229227374253, −0.64718828989727598030993658141, −0.17363926125865957323674504765, 0.17363926125865957323674504765, 0.64718828989727598030993658141, 0.72368989513177506229227374253, 0.927190181250870571167176206248, 1.38446150759953836530121938653, 1.80284805721405326829505587038, 1.93614479238635172144932758884, 1.94836447212783504927597628299, 2.51448665315435471446402228183, 2.61137784043725239779945589118, 2.61920292831551753897990457283, 2.77105187206083539127962618620, 3.09488433174261509832197533352, 3.45061999192101978908655795775, 3.58645152038127637363863219682, 3.64963271525450533347696601410, 3.96535071998907838090486600450, 4.15082971204870365831830797329, 4.21403388844508944825607083619, 4.48822459196255333159797730374, 4.89352609033869587113095039843, 4.93740213913496464232968464666, 4.96535600581706233654654978046, 5.34032789337258037949447130142, 5.43129413578133342196283885383

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