Properties

Label 8-1200e4-1.1-c2e4-0-9
Degree $8$
Conductor $2.074\times 10^{12}$
Sign $1$
Analytic cond. $1.14304\times 10^{6}$
Root an. cond. $5.71818$
Motivic weight $2$
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  − 24·11-s − 100·31-s − 240·41-s − 148·61-s − 528·71-s − 9·81-s + 384·101-s − 124·121-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 + ⋯
L(s)  = 1  − 2.18·11-s − 3.22·31-s − 5.85·41-s − 2.42·61-s − 7.43·71-s − 1/9·81-s + 3.80·101-s − 1.02·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.14304\times 10^{6}\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1116036959\)
\(L(\frac12)\) \(\approx\) \(0.1116036959\)
\(L(2)\) 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$C_2^2$ \( 1 + p^{2} T^{4} \)
5 \( 1 \)
good7$C_2^3$ \( 1 - 4273 T^{4} + p^{8} T^{8} \)
11$C_2$ \( ( 1 + 6 T + p^{2} T^{2} )^{4} \)
13$C_2^3$ \( 1 + 39599 T^{4} + p^{8} T^{8} \)
17$C_2^3$ \( 1 - 166942 T^{4} + p^{8} T^{8} \)
19$C_2^2$ \( ( 1 - 193 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 14882 T^{4} + p^{8} T^{8} \)
29$C_2^2$ \( ( 1 - 1646 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 25 T + p^{2} T^{2} )^{4} \)
37$C_2^3$ \( 1 - 1382878 T^{4} + p^{8} T^{8} \)
41$C_2$ \( ( 1 + 60 T + p^{2} T^{2} )^{4} \)
43$C_2^3$ \( 1 + 5447423 T^{4} + p^{8} T^{8} \)
47$C_2^3$ \( 1 + 8816738 T^{4} + p^{8} T^{8} \)
53$C_2^3$ \( 1 + 3737762 T^{4} + p^{8} T^{8} \)
59$C_2^2$ \( ( 1 - 6638 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 37 T + p^{2} T^{2} )^{4} \)
67$C_2^3$ \( 1 + 18296783 T^{4} + p^{8} T^{8} \)
71$C_2$ \( ( 1 + 132 T + p^{2} T^{2} )^{4} \)
73$C_2^3$ \( 1 + 32657282 T^{4} + p^{8} T^{8} \)
79$C_2^2$ \( ( 1 - 12382 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^3$ \( 1 + 94586114 T^{4} + p^{8} T^{8} \)
89$C_2^2$ \( ( 1 + 1582 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^3$ \( 1 + 137471663 T^{4} + p^{8} T^{8} \)
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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

−6.79741348571502496470554918852, −6.66113176017830216752410575509, −6.43459340112788763763774558666, −5.89934409882992625476250111462, −5.70300250790408679145084386386, −5.66769022259581842064708802577, −5.62609794392989111016584311681, −5.03627164117767721635661846612, −5.03577287820607549354276278763, −4.78298395808719744222346041661, −4.62181871224529596706046240867, −4.36229222984968919845925208193, −3.88268279480830768077915357551, −3.57118014050987901482664767492, −3.55617103678790265535894352613, −3.06236455875974967860217189171, −2.96562120442314982147003798158, −2.89311608303155315254284281054, −2.31663252258748454664370920522, −1.93857473759009628059302047504, −1.77247444014346842780790264651, −1.47687635041484351081835433437, −1.29253212873210595414655590046, −0.17788097427439186819908837285, −0.15835584201014089970708490302, 0.15835584201014089970708490302, 0.17788097427439186819908837285, 1.29253212873210595414655590046, 1.47687635041484351081835433437, 1.77247444014346842780790264651, 1.93857473759009628059302047504, 2.31663252258748454664370920522, 2.89311608303155315254284281054, 2.96562120442314982147003798158, 3.06236455875974967860217189171, 3.55617103678790265535894352613, 3.57118014050987901482664767492, 3.88268279480830768077915357551, 4.36229222984968919845925208193, 4.62181871224529596706046240867, 4.78298395808719744222346041661, 5.03577287820607549354276278763, 5.03627164117767721635661846612, 5.62609794392989111016584311681, 5.66769022259581842064708802577, 5.70300250790408679145084386386, 5.89934409882992625476250111462, 6.43459340112788763763774558666, 6.66113176017830216752410575509, 6.79741348571502496470554918852

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