Properties

Label 8-768e4-1.1-c2e4-0-9
Degree $8$
Conductor $347892350976$
Sign $1$
Analytic cond. $191771.$
Root an. cond. $4.57454$
Motivic weight $2$
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  + 16·7-s + 10·9-s + 36·25-s + 16·31-s − 36·49-s + 160·63-s + 24·73-s − 496·79-s + 19·81-s + 472·97-s − 432·103-s + 340·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 452·169-s + 173-s + 576·175-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 16/7·7-s + 10/9·9-s + 1.43·25-s + 0.516·31-s − 0.734·49-s + 2.53·63-s + 0.328·73-s − 6.27·79-s + 0.234·81-s + 4.86·97-s − 4.19·103-s + 2.80·121-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 − 2.67·169-s + 0.00578·173-s + 3.29·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(191771.\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 3^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.361541270\)
\(L(\frac12)\) \(\approx\) \(3.361541270\)
\(L(2)\) 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$C_2^2$ \( 1 - 10 T^{2} + p^{4} T^{4} \)
good5$C_2^2$ \( ( 1 - 18 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p^{2} T^{2} )^{4} \)
11$C_2^2$ \( ( 1 - 170 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 226 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 354 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 694 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 162 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 1394 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p^{2} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 2466 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 3670 T^{2} + p^{4} T^{4} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
53$C_2^2$ \( ( 1 - 3026 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 4650 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 1630 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 6710 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 110 T + p^{2} T^{2} )^{2}( 1 + 110 T + p^{2} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p^{2} T^{2} )^{4} \)
79$C_2$ \( ( 1 + 124 T + p^{2} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 13770 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 4866 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 118 T + p^{2} T^{2} )^{4} \)
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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

−7.15689242114164880538189955036, −7.03642459216801826422852936159, −6.97969467622314810166845736684, −6.49741959741611075128797145913, −6.23648214274143357822565483643, −6.00022444133876336051597924029, −5.93535516248222915796441825173, −5.39282001499653927473633203524, −5.17419844384268338855875253491, −5.09343750095875195517073232288, −4.66362481221951809623032496380, −4.57748419707115684093989836501, −4.45640400140106467124871703830, −4.28182366932222526987601089678, −3.71769348938364777753852340830, −3.44303437590024790278593847159, −3.34248041376167528347105109156, −2.71612446419950499375836278774, −2.56497221087536445687691213890, −2.29286587066600443685351474201, −1.63188128780547973630919131323, −1.46559481316294171931335995312, −1.41007863511449309301620003212, −1.01372184291647032389329540319, −0.25522889994650199847570984614, 0.25522889994650199847570984614, 1.01372184291647032389329540319, 1.41007863511449309301620003212, 1.46559481316294171931335995312, 1.63188128780547973630919131323, 2.29286587066600443685351474201, 2.56497221087536445687691213890, 2.71612446419950499375836278774, 3.34248041376167528347105109156, 3.44303437590024790278593847159, 3.71769348938364777753852340830, 4.28182366932222526987601089678, 4.45640400140106467124871703830, 4.57748419707115684093989836501, 4.66362481221951809623032496380, 5.09343750095875195517073232288, 5.17419844384268338855875253491, 5.39282001499653927473633203524, 5.93535516248222915796441825173, 6.00022444133876336051597924029, 6.23648214274143357822565483643, 6.49741959741611075128797145913, 6.97969467622314810166845736684, 7.03642459216801826422852936159, 7.15689242114164880538189955036

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