Properties

Label 8-768e4-1.1-c3e4-0-9
Degree $8$
Conductor $347892350976$
Sign $1$
Analytic cond. $4.21608\times 10^{6}$
Root an. cond. $6.73152$
Motivic weight $3$
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  − 46·9-s + 500·25-s + 1.37e3·49-s + 1.72e3·73-s + 1.38e3·81-s + 7.64e3·97-s + 4.67e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8.78e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 2.30e4·225-s + ⋯
L(s)  = 1  − 1.70·9-s + 4·25-s + 4·49-s + 2.75·73-s + 1.90·81-s + 7.99·97-s + 3.51·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 4·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s − 6.81·225-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(4.21608\times 10^{6}\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 3^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(6.202457961\)
\(L(\frac12)\) \(\approx\) \(6.202457961\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 + 46 T^{2} + p^{6} T^{4} \)
good5$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
7$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
11$C_2^2$ \( ( 1 - 2338 T^{2} + p^{6} T^{4} )^{2} \)
13$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
17$C_2$ \( ( 1 - 90 T + p^{3} T^{2} )^{2}( 1 + 90 T + p^{3} T^{2} )^{2} \)
19$C_2^2$ \( ( 1 - 2482 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
29$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
31$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
37$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
41$C_2$ \( ( 1 - 522 T + p^{3} T^{2} )^{2}( 1 + 522 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( ( 1 - 74914 T^{2} + p^{6} T^{4} )^{2} \)
47$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
53$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + 304958 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 596626 T^{2} + p^{6} T^{4} )^{2} \)
71$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
73$C_2$ \( ( 1 - 430 T + p^{3} T^{2} )^{4} \)
79$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 678926 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 1026 T + p^{3} T^{2} )^{2}( 1 + 1026 T + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 1910 T + p^{3} T^{2} )^{4} \)
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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

−7.21521096447116099154626349203, −6.53846826699607259004989635913, −6.51456621734540357874192327613, −6.40811559928550661535380054751, −6.15424724215457557259438207452, −5.82523191190479630978365329970, −5.49747181202334940224917698809, −5.27874731163063849920260060483, −5.26807208076078694667529097722, −4.87020001310443470226152410415, −4.58624731374813109069251631727, −4.47386191744277633701306592946, −4.18164382761516558379980819551, −3.54707222972038038046170602756, −3.38623084379226616559200859771, −3.36555950349182472985105137782, −3.08840463530599324690586199635, −2.49267657040859462314327459796, −2.38750300682225295132425016367, −2.27803714648831478749855355656, −1.85664890348767995639481792744, −1.08671366896677629974371244279, −0.811632390146552904316313346518, −0.78578255270400873106345616900, −0.38281618218239628653914413800, 0.38281618218239628653914413800, 0.78578255270400873106345616900, 0.811632390146552904316313346518, 1.08671366896677629974371244279, 1.85664890348767995639481792744, 2.27803714648831478749855355656, 2.38750300682225295132425016367, 2.49267657040859462314327459796, 3.08840463530599324690586199635, 3.36555950349182472985105137782, 3.38623084379226616559200859771, 3.54707222972038038046170602756, 4.18164382761516558379980819551, 4.47386191744277633701306592946, 4.58624731374813109069251631727, 4.87020001310443470226152410415, 5.26807208076078694667529097722, 5.27874731163063849920260060483, 5.49747181202334940224917698809, 5.82523191190479630978365329970, 6.15424724215457557259438207452, 6.40811559928550661535380054751, 6.51456621734540357874192327613, 6.53846826699607259004989635913, 7.21521096447116099154626349203

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