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Properties

Label 10-1459e5-1459.1458-c0e5-0-0
Degree $10$
Conductor $6.611\times 10^{15}$
Sign $1$
Analytic cond. $0.204673$
Root an. cond. $0.853308$
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  + 5·4-s − 5-s − 7-s + 5·9-s − 11-s − 13-s + 15·16-s − 19-s − 5·20-s − 23-s − 5·28-s − 29-s + 35-s + 25·36-s − 37-s − 5·44-s − 5·45-s − 5·52-s − 53-s + 55-s − 59-s − 61-s − 5·63-s + 35·64-s + 65-s − 67-s − 71-s + ⋯
L(s)  = 1  + 5·4-s − 5-s − 7-s + 5·9-s − 11-s − 13-s + 15·16-s − 19-s − 5·20-s − 23-s − 5·28-s − 29-s + 35-s + 25·36-s − 37-s − 5·44-s − 5·45-s − 5·52-s − 53-s + 55-s − 59-s − 61-s − 5·63-s + 35·64-s + 65-s − 67-s − 71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1459^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1459^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(1459^{5}\)
Sign: $1$
Analytic conductor: \(0.204673\)
Root analytic conductor: \(0.853308\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1459} (1458, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 1459^{5} ,\ ( \ : 0, 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.761162217\)
\(L(\frac12)\) \(\approx\) \(4.761162217\)
\(L(1)\) 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)$
bad1459$C_1$ \( ( 1 - T )^{5} \)
good2$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
5$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
7$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
11$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
13$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
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$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
53$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
59$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
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$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
73$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
79$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
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

−6.09276099873405151750805172394, −5.94101000008087934883674537216, −5.63937762556765577931566583693, −5.60205295833585044926683003475, −5.32822555545982769086593664428, −5.08973571457308550394774693966, −4.68505409187992866795331309771, −4.59829353825087890721434429735, −4.59234024232345296669310638244, −4.23663891909322990795871613560, −3.84754744101563315291057470563, −3.72534105006577340099337657606, −3.66580329762862007048212441943, −3.65104377361790214504450618329, −3.24140210710881068019706895608, −2.93104562762486041812624544789, −2.76467437628934669075834320911, −2.61719054357699101594761932514, −2.25810280770396756658698464124, −2.25687694471343747855354459430, −1.74592228645673644367640977226, −1.68803442696146212275327765869, −1.62869784493136765686045135249, −1.26751441037037164555389979434, −1.19603171172213874079885864206, 1.19603171172213874079885864206, 1.26751441037037164555389979434, 1.62869784493136765686045135249, 1.68803442696146212275327765869, 1.74592228645673644367640977226, 2.25687694471343747855354459430, 2.25810280770396756658698464124, 2.61719054357699101594761932514, 2.76467437628934669075834320911, 2.93104562762486041812624544789, 3.24140210710881068019706895608, 3.65104377361790214504450618329, 3.66580329762862007048212441943, 3.72534105006577340099337657606, 3.84754744101563315291057470563, 4.23663891909322990795871613560, 4.59234024232345296669310638244, 4.59829353825087890721434429735, 4.68505409187992866795331309771, 5.08973571457308550394774693966, 5.32822555545982769086593664428, 5.60205295833585044926683003475, 5.63937762556765577931566583693, 5.94101000008087934883674537216, 6.09276099873405151750805172394

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