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Properties

Label 8-72e4-1.1-c1e4-0-3
Degree $8$
Conductor $26873856$
Sign $1$
Analytic cond. $0.109254$
Root an. cond. $0.758236$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·4-s + 3·9-s − 18·11-s + 4·12-s − 4·19-s + 10·25-s + 10·27-s − 36·33-s + 6·36-s + 18·41-s − 10·43-s − 36·44-s − 14·49-s − 8·57-s + 18·59-s − 8·64-s + 14·67-s − 4·73-s + 20·75-s − 8·76-s + 20·81-s − 10·97-s − 54·99-s + 20·100-s + 20·108-s + 169·121-s + ⋯
L(s)  = 1  + 1.15·3-s + 4-s + 9-s − 5.42·11-s + 1.15·12-s − 0.917·19-s + 2·25-s + 1.92·27-s − 6.26·33-s + 36-s + 2.81·41-s − 1.52·43-s − 5.42·44-s − 2·49-s − 1.05·57-s + 2.34·59-s − 64-s + 1.71·67-s − 0.468·73-s + 2.30·75-s − 0.917·76-s + 20/9·81-s − 1.01·97-s − 5.42·99-s + 2·100-s + 1.92·108-s + 15.3·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(26873856\)    =    \(2^{12} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(0.109254\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 26873856,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9940479530\)
\(L(\frac12)\) \(\approx\) \(0.9940479530\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \)
good5$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
11$C_2$$\times$$C_2^2$ \( ( 1 + 6 T + p T^{2} )^{2}( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
13$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
19$C_2^2$ \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{4} \)
41$C_2$$\times$$C_2^2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \)
43$C_2$$\times$$C_2^2$ \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \)
47$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$C_2$$\times$$C_2^2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} ) \)
61$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$$\times$$C_2^2$ \( ( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \)
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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

−10.82257133950445822006569975190, −10.65776290524712935405876278964, −10.32980201828229267574607016979, −10.12341874679207186463613675070, −9.918566536362943500092950394430, −9.487939876288600041548634650951, −8.976847986167529653157356874982, −8.542133080835188017557302649565, −8.323805465493111519116727905457, −8.193271439978103121820558346488, −7.86177047040072258534126108663, −7.51439919872580222149725574821, −7.33584879936360968511900110449, −6.88674705314451405041949624189, −6.56921485253876360296646337170, −6.10263812145192357411800228002, −5.54245330779234409972191726848, −5.14872745705443002593593979642, −4.91148107712993364417179104961, −4.73303866172930221653669861932, −3.89150805359828389086771997983, −3.00141070934747919454022704351, −2.87003400110696441359977158822, −2.46497567617430314622651811016, −2.26572919897296598094005540027, 2.26572919897296598094005540027, 2.46497567617430314622651811016, 2.87003400110696441359977158822, 3.00141070934747919454022704351, 3.89150805359828389086771997983, 4.73303866172930221653669861932, 4.91148107712993364417179104961, 5.14872745705443002593593979642, 5.54245330779234409972191726848, 6.10263812145192357411800228002, 6.56921485253876360296646337170, 6.88674705314451405041949624189, 7.33584879936360968511900110449, 7.51439919872580222149725574821, 7.86177047040072258534126108663, 8.193271439978103121820558346488, 8.323805465493111519116727905457, 8.542133080835188017557302649565, 8.976847986167529653157356874982, 9.487939876288600041548634650951, 9.918566536362943500092950394430, 10.12341874679207186463613675070, 10.32980201828229267574607016979, 10.65776290524712935405876278964, 10.82257133950445822006569975190

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