Properties

Label 10-591e5-591.590-c0e5-0-0
Degree $10$
Conductor $7.210\times 10^{13}$
Sign $1$
Analytic cond. $0.00223214$
Root an. cond. $0.543090$
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 − 7-s + 15·9-s + 10-s + 11-s − 14-s − 5·15-s + 17-s + 15·18-s − 19-s + 5·21-s + 22-s − 35·27-s − 5·30-s − 5·33-s + 34-s − 35-s − 37-s − 38-s + 5·42-s − 43-s + 15·45-s − 5·51-s − 35·54-s + ⋯
L(s)  = 1  + 2-s − 5·3-s + 5-s − 5·6-s − 7-s + 15·9-s + 10-s + 11-s − 14-s − 5·15-s + 17-s + 15·18-s − 19-s + 5·21-s + 22-s − 35·27-s − 5·30-s − 5·33-s + 34-s − 35-s − 37-s − 38-s + 5·42-s − 43-s + 15·45-s − 5·51-s − 35·54-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 197^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 197^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{5} \cdot 197^{5}\)
Sign: $1$
Analytic conductor: \(0.00223214\)
Root analytic conductor: \(0.543090\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{591} (590, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 3^{5} \cdot 197^{5} ,\ ( \ : 0, 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1691454834\)
\(L(\frac12)\) \(\approx\) \(0.1691454834\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{5} \)
197$C_1$ \( ( 1 + T )^{5} \)
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$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
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$C_{10}$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
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$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
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$C_{10}$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
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

−6.66237059500660415356588547842, −6.30426624176626154788582360319, −6.22552313158709465908198716243, −6.19623899487199539124863349461, −6.12329184503179824230534581395, −5.76896124072112502200366667927, −5.43701637745393464281762277777, −5.39087448232256428587576153025, −5.34653707680592805465235910286, −5.13051133890941151135768396585, −4.89096287639849296897434723588, −4.54135777902142320172860263600, −4.50655861583626393508981776065, −4.28591092828602881799368338735, −4.13106980047591615744440387510, −3.92019783737823609810945523924, −3.47580961333898468040158924752, −3.43098296892257340637838362892, −3.21192069259822105248431137681, −2.27185101973434999401567118730, −2.24500824390988576284965548980, −1.71661130588285073503029844428, −1.42980776969111633705730394341, −1.33022535577687141478853731364, −0.65344816078370669748145330666, 0.65344816078370669748145330666, 1.33022535577687141478853731364, 1.42980776969111633705730394341, 1.71661130588285073503029844428, 2.24500824390988576284965548980, 2.27185101973434999401567118730, 3.21192069259822105248431137681, 3.43098296892257340637838362892, 3.47580961333898468040158924752, 3.92019783737823609810945523924, 4.13106980047591615744440387510, 4.28591092828602881799368338735, 4.50655861583626393508981776065, 4.54135777902142320172860263600, 4.89096287639849296897434723588, 5.13051133890941151135768396585, 5.34653707680592805465235910286, 5.39087448232256428587576153025, 5.43701637745393464281762277777, 5.76896124072112502200366667927, 6.12329184503179824230534581395, 6.19623899487199539124863349461, 6.22552313158709465908198716243, 6.30426624176626154788582360319, 6.66237059500660415356588547842

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