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Properties

Label 8-840e4-1.1-c0e4-0-2
Degree $8$
Conductor $497871360000$
Sign $1$
Analytic cond. $0.0308848$
Root an. cond. $0.647467$
Motivic weight $0$
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  − 16-s − 2·49-s + 4·73-s − 81-s + 4·97-s + 4·103-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 + 233-s + ⋯
L(s)  = 1  − 16-s − 2·49-s + 4·73-s − 81-s + 4·97-s + 4·103-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 + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.0308848\)
Root analytic conductor: \(0.647467\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7313208405\)
\(L(\frac12)\) \(\approx\) \(0.7313208405\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
3$C_2^2$ \( 1 + T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
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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

−7.60830839848579287449144212332, −7.34791635546242617611968444033, −7.09625253732592273389670699645, −6.92531020987349327793986997360, −6.53929113813073356187731439951, −6.36791357826312588518274170699, −6.21183307845880297793203296716, −6.01737394771442801533937004071, −5.84656096314949708592075285408, −5.35456511869911798075528263888, −5.08946748154639789676236287643, −4.94618793059937242988507428077, −4.75427825159890925897174862724, −4.50221113700582711880768876478, −4.36144758484577729371650016159, −3.74190083982064994031321079220, −3.55255530132479921596922979544, −3.43209531423189077168190502280, −3.29726981972386314402310601911, −2.63967026322946297514642594706, −2.29174084579632205970616944395, −2.25844481267753838998040089743, −1.86498799025918509090511760598, −1.34108280495155758038607577854, −0.808146649981180763141789924352, 0.808146649981180763141789924352, 1.34108280495155758038607577854, 1.86498799025918509090511760598, 2.25844481267753838998040089743, 2.29174084579632205970616944395, 2.63967026322946297514642594706, 3.29726981972386314402310601911, 3.43209531423189077168190502280, 3.55255530132479921596922979544, 3.74190083982064994031321079220, 4.36144758484577729371650016159, 4.50221113700582711880768876478, 4.75427825159890925897174862724, 4.94618793059937242988507428077, 5.08946748154639789676236287643, 5.35456511869911798075528263888, 5.84656096314949708592075285408, 6.01737394771442801533937004071, 6.21183307845880297793203296716, 6.36791357826312588518274170699, 6.53929113813073356187731439951, 6.92531020987349327793986997360, 7.09625253732592273389670699645, 7.34791635546242617611968444033, 7.60830839848579287449144212332

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