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Properties

Label 10-167e5-167.166-c0e5-0-0
Degree $10$
Conductor $129891985607$
Sign $1$
Analytic cond. $4.02131\times 10^{-6}$
Root an. cond. $0.288693$
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  − 2-s − 3-s + 6-s − 7-s − 11-s + 14-s − 19-s + 21-s + 22-s + 5·25-s − 29-s − 31-s + 33-s + 38-s − 42-s − 47-s − 5·50-s + 57-s + 58-s − 61-s + 62-s − 66-s − 5·75-s + 77-s + 87-s − 89-s + 93-s + ⋯
L(s)  = 1  − 2-s − 3-s + 6-s − 7-s − 11-s + 14-s − 19-s + 21-s + 22-s + 5·25-s − 29-s − 31-s + 33-s + 38-s − 42-s − 47-s − 5·50-s + 57-s + 58-s − 61-s + 62-s − 66-s − 5·75-s + 77-s + 87-s − 89-s + 93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(167^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(167^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(167^{5}\)
Sign: $1$
Analytic conductor: \(4.02131\times 10^{-6}\)
Root analytic conductor: \(0.288693\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{167} (166, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 167^{5} ,\ ( \ : 0, 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04746970378\)
\(L(\frac12)\) \(\approx\) \(0.04746970378\)
\(L(1)\) 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)$
bad167$C_1$ \( ( 1 - T )^{5} \)
good2$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
3$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
7$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
11$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
19$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
29$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
31$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
47$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
61$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
89$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
97$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.333795454308991850634630192916, −8.213248461003269160838496361340, −8.071189840663236759526990153770, −7.73916040331477618831346896949, −7.27492032693309654900391920139, −7.10800749405324563640789273519, −6.97522263091615297464090375854, −6.89653435720459883009277436790, −6.36230016194657339559253630949, −6.33288394360772769794385190186, −6.23245137893618395901473074444, −5.59369039149323546264449937798, −5.53971189531068120511188939291, −5.30109483791642467558661579351, −5.11393199942617971707444865634, −4.63446809423050619209573506866, −4.59175755017292301723831102547, −4.27419318822733967071565800275, −3.70752451965268053902922861373, −3.44327788271159140430101767030, −3.09426105654948352496101562792, −2.71831488467609106969659188153, −2.67413738571413214883570764077, −1.91443004313293629718250194665, −1.25497611148562736482137054417, 1.25497611148562736482137054417, 1.91443004313293629718250194665, 2.67413738571413214883570764077, 2.71831488467609106969659188153, 3.09426105654948352496101562792, 3.44327788271159140430101767030, 3.70752451965268053902922861373, 4.27419318822733967071565800275, 4.59175755017292301723831102547, 4.63446809423050619209573506866, 5.11393199942617971707444865634, 5.30109483791642467558661579351, 5.53971189531068120511188939291, 5.59369039149323546264449937798, 6.23245137893618395901473074444, 6.33288394360772769794385190186, 6.36230016194657339559253630949, 6.89653435720459883009277436790, 6.97522263091615297464090375854, 7.10800749405324563640789273519, 7.27492032693309654900391920139, 7.73916040331477618831346896949, 8.071189840663236759526990153770, 8.213248461003269160838496361340, 8.333795454308991850634630192916

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