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Properties

Label 10-1391e5-1391.1390-c0e5-0-0
Degree $10$
Conductor $5.208\times 10^{15}$
Sign $1$
Analytic cond. $0.161220$
Root an. cond. $0.833186$
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 − 3-s − 5-s + 6-s − 7-s + 10-s + 5·13-s + 14-s + 15-s + 21-s − 23-s − 5·26-s − 29-s − 30-s − 31-s + 35-s − 5·39-s − 42-s + 46-s − 53-s + 58-s − 59-s − 61-s + 62-s − 5·65-s + 10·67-s + 69-s + ⋯
L(s)  = 1  − 2-s − 3-s − 5-s + 6-s − 7-s + 10-s + 5·13-s + 14-s + 15-s + 21-s − 23-s − 5·26-s − 29-s − 30-s − 31-s + 35-s − 5·39-s − 42-s + 46-s − 53-s + 58-s − 59-s − 61-s + 62-s − 5·65-s + 10·67-s + 69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{5} \cdot 107^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{5} \cdot 107^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(13^{5} \cdot 107^{5}\)
Sign: $1$
Analytic conductor: \(0.161220\)
Root analytic conductor: \(0.833186\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1391} (1390, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 13^{5} \cdot 107^{5} ,\ ( \ : 0, 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2534963871\)
\(L(\frac12)\) \(\approx\) \(0.2534963871\)
\(L(1)\) 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)$
bad13$C_1$ \( ( 1 - T )^{5} \)
107$C_1$ \( ( 1 - T )^{5} \)
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$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
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_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_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_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$C_1$ \( ( 1 - T )^{10} \)
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

−6.03035722057000624732805104263, −5.74421423522075212833854782337, −5.68477256859220785560490203834, −5.66936013038205933519966964300, −5.42797961625771839501008166475, −5.30089231682600966514056164875, −4.80931462055391376459994359468, −4.69013218383786105329779471889, −4.45860175311463611312653891343, −4.28247948538775010053888940755, −3.90536986496050451816721707360, −3.78235732439260252454545227323, −3.76668699152553737508788606428, −3.72893133190105205641897812865, −3.33463580037704733252122090261, −3.22400162619791631851078833772, −2.96467420247758789854774133416, −2.94284755241177636739768672321, −2.05985520246771546725900796194, −2.03793241121578481436445139615, −1.89679864428696718607776641637, −1.67002117838606277392959023120, −1.02010850813908948224464320676, −0.902773190750548242847859588893, −0.56321720549631269908513583905, 0.56321720549631269908513583905, 0.902773190750548242847859588893, 1.02010850813908948224464320676, 1.67002117838606277392959023120, 1.89679864428696718607776641637, 2.03793241121578481436445139615, 2.05985520246771546725900796194, 2.94284755241177636739768672321, 2.96467420247758789854774133416, 3.22400162619791631851078833772, 3.33463580037704733252122090261, 3.72893133190105205641897812865, 3.76668699152553737508788606428, 3.78235732439260252454545227323, 3.90536986496050451816721707360, 4.28247948538775010053888940755, 4.45860175311463611312653891343, 4.69013218383786105329779471889, 4.80931462055391376459994359468, 5.30089231682600966514056164875, 5.42797961625771839501008166475, 5.66936013038205933519966964300, 5.68477256859220785560490203834, 5.74421423522075212833854782337, 6.03035722057000624732805104263

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