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Properties

Label 8-3536e4-1.1-c0e4-0-4
Degree $8$
Conductor $1.563\times 10^{14}$
Sign $1$
Analytic cond. $9.69789$
Root an. cond. $1.32841$
Motivic weight $0$
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  + 4·7-s + 4·11-s + 4·17-s − 4·19-s − 4·31-s + 10·49-s − 4·59-s + 4·61-s + 16·77-s + 16·119-s + 10·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 16·187-s + 191-s + ⋯
L(s)  = 1  + 4·7-s + 4·11-s + 4·17-s − 4·19-s − 4·31-s + 10·49-s − 4·59-s + 4·61-s + 16·77-s + 16·119-s + 10·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 16·187-s + 191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 13^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(9.69789\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 13^{4} \cdot 17^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.308413626\)
\(L(\frac12)\) \(\approx\) \(5.308413626\)
\(L(1)\) 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 \( 1 \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_1$ \( ( 1 - T )^{4} \)
good3$C_4\times C_2$ \( 1 + T^{8} \)
5$C_4\times C_2$ \( 1 + T^{8} \)
7$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
11$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
23$C_4\times C_2$ \( 1 + T^{8} \)
29$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
31$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
37$C_4\times C_2$ \( 1 + T^{8} \)
41$C_4\times C_2$ \( 1 + T^{8} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
61$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
73$C_4\times C_2$ \( 1 + T^{8} \)
79$C_4\times C_2$ \( 1 + T^{8} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_4\times C_2$ \( 1 + T^{8} \)
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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

−6.28610461489292075020125038768, −5.82554737287717930316175855053, −5.78441157216822722204420176419, −5.65787178085795478919788231071, −5.64475842147368706365379699102, −5.16986754801424293037742594531, −5.00005247464692354221608421764, −4.90469592713179630597914213167, −4.61468409084732001912786750308, −4.34518774513205749685327696054, −4.17407687211064710244140711468, −4.01296245533721836591930632368, −3.95331049723018105715266706884, −3.71573413621836351189677079384, −3.42380600533520763483895202614, −3.36890111900793522041258081456, −2.94197818581123644409714947543, −2.37379941266993658082225485660, −2.05049512282581678332565981703, −1.99133899138846129487392598040, −1.89165184224154220249585687522, −1.56631331079462135252561265614, −1.29375064779909553645932792265, −1.10688841840883569551758454391, −0.987063898612726495354128234983, 0.987063898612726495354128234983, 1.10688841840883569551758454391, 1.29375064779909553645932792265, 1.56631331079462135252561265614, 1.89165184224154220249585687522, 1.99133899138846129487392598040, 2.05049512282581678332565981703, 2.37379941266993658082225485660, 2.94197818581123644409714947543, 3.36890111900793522041258081456, 3.42380600533520763483895202614, 3.71573413621836351189677079384, 3.95331049723018105715266706884, 4.01296245533721836591930632368, 4.17407687211064710244140711468, 4.34518774513205749685327696054, 4.61468409084732001912786750308, 4.90469592713179630597914213167, 5.00005247464692354221608421764, 5.16986754801424293037742594531, 5.64475842147368706365379699102, 5.65787178085795478919788231071, 5.78441157216822722204420176419, 5.82554737287717930316175855053, 6.28610461489292075020125038768

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