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Properties

Label 10-879e5-879.878-c0e5-0-1
Degree $10$
Conductor $5.247\times 10^{14}$
Sign $1$
Analytic cond. $0.0162453$
Root an. cond. $0.662327$
Motivic weight $0$
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  − 2-s + 5·3-s − 5-s − 5·6-s + 15·9-s + 10-s − 11-s − 5·15-s − 15·18-s + 22-s − 23-s + 35·27-s − 29-s + 5·30-s − 31-s − 5·33-s − 37-s − 41-s − 43-s − 15·45-s + 46-s − 47-s + 5·49-s − 35·54-s + 55-s + 58-s − 61-s + ⋯
L(s)  = 1  − 2-s + 5·3-s − 5-s − 5·6-s + 15·9-s + 10-s − 11-s − 5·15-s − 15·18-s + 22-s − 23-s + 35·27-s − 29-s + 5·30-s − 31-s − 5·33-s − 37-s − 41-s − 43-s − 15·45-s + 46-s − 47-s + 5·49-s − 35·54-s + 55-s + 58-s − 61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 293^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 293^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{5} \cdot 293^{5}\)
Sign: $1$
Analytic conductor: \(0.0162453\)
Root analytic conductor: \(0.662327\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{879} (878, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 3^{5} \cdot 293^{5} ,\ ( \ : 0, 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.112479657\)
\(L(\frac12)\) \(\approx\) \(2.112479657\)
\(L(1)\) 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)$
bad3$C_1$ \( ( 1 - T )^{5} \)
293$C_1$ \( ( 1 - T )^{5} \)
good2$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
5$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
23$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
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$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
41$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
43$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
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$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
73$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
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

−6.68879604512109589377866523040, −6.56554382967175735908728641218, −6.01454171264605357957964889389, −5.82423765913002132062220951374, −5.66095895237944728570575069259, −5.35467186882321865667214040824, −5.13674625917403451430643524292, −4.77820657218015970397412088605, −4.67852953794577333339025291011, −4.52970321777187060903065463332, −3.99104812474157171121315699757, −3.98139019121652098469235905744, −3.97841280294176679218019318850, −3.90995174456612058813909072952, −3.39883597495345988953418837815, −3.35285606028953429361807890432, −3.16671189263066043441787656173, −2.80871385947112733631143316968, −2.65161594596023544126897136666, −2.38096744280373541079599630166, −2.36449039140924332708756883058, −1.82652491670097267516921567940, −1.72628388281442364501142299747, −1.50163835474436434735355316051, −1.10230737811382492112475340453, 1.10230737811382492112475340453, 1.50163835474436434735355316051, 1.72628388281442364501142299747, 1.82652491670097267516921567940, 2.36449039140924332708756883058, 2.38096744280373541079599630166, 2.65161594596023544126897136666, 2.80871385947112733631143316968, 3.16671189263066043441787656173, 3.35285606028953429361807890432, 3.39883597495345988953418837815, 3.90995174456612058813909072952, 3.97841280294176679218019318850, 3.98139019121652098469235905744, 3.99104812474157171121315699757, 4.52970321777187060903065463332, 4.67852953794577333339025291011, 4.77820657218015970397412088605, 5.13674625917403451430643524292, 5.35467186882321865667214040824, 5.66095895237944728570575069259, 5.82423765913002132062220951374, 6.01454171264605357957964889389, 6.56554382967175735908728641218, 6.68879604512109589377866523040

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