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Properties

Label 10-1076e5-1076.1075-c0e5-0-1
Degree $10$
Conductor $1.442\times 10^{15}$
Sign $1$
Analytic cond. $0.0446526$
Root an. cond. $0.732798$
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  + 5·2-s − 3-s + 15·4-s − 5-s − 5·6-s − 7-s + 35·8-s − 5·10-s − 15·12-s − 13-s − 5·14-s + 15-s + 70·16-s − 19-s − 15·20-s + 21-s − 35·24-s − 5·26-s − 15·28-s + 5·30-s − 31-s + 126·32-s + 35-s − 37-s − 5·38-s + 39-s − 35·40-s + ⋯
L(s)  = 1  + 5·2-s − 3-s + 15·4-s − 5-s − 5·6-s − 7-s + 35·8-s − 5·10-s − 15·12-s − 13-s − 5·14-s + 15-s + 70·16-s − 19-s − 15·20-s + 21-s − 35·24-s − 5·26-s − 15·28-s + 5·30-s − 31-s + 126·32-s + 35-s − 37-s − 5·38-s + 39-s − 35·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 269^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 269^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{10} \cdot 269^{5}\)
Sign: $1$
Analytic conductor: \(0.0446526\)
Root analytic conductor: \(0.732798\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1076} (1075, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{10} \cdot 269^{5} ,\ ( \ : 0, 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(13.70762966\)
\(L(\frac12)\) \(\approx\) \(13.70762966\)
\(L(1)\) 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)$
bad2$C_1$ \( ( 1 - T )^{5} \)
269$C_1$ \( ( 1 - T )^{5} \)
good3$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
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$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
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_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$$\times$$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$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
83$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
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

−5.98674601227753163190618742982, −5.95277900894735785534928954561, −5.73677781078729174526430216034, −5.61022646993261522194644557636, −5.45524109768114353801183822321, −5.27446975984209364591846858169, −5.09352510912561169219676630219, −4.71411978641204466280725551773, −4.66038776736180582045546907134, −4.45713504168616668148773390167, −4.41791254467034669397060970649, −4.33164030601427182734310720479, −3.90856056598605655124806739162, −3.77344807102863788791801830454, −3.53476416136056540545959072080, −3.23587462167296875994030557376, −3.22638474014413499174007977989, −3.12639946681664679668092546249, −2.84279748723506265971109581613, −2.52776539572239009562812309498, −2.33209857940370992511706074842, −1.85838305487200298685101373606, −1.78487326180741643092626540059, −1.71132706113428793846777337338, −1.10529274823556114725015056687, 1.10529274823556114725015056687, 1.71132706113428793846777337338, 1.78487326180741643092626540059, 1.85838305487200298685101373606, 2.33209857940370992511706074842, 2.52776539572239009562812309498, 2.84279748723506265971109581613, 3.12639946681664679668092546249, 3.22638474014413499174007977989, 3.23587462167296875994030557376, 3.53476416136056540545959072080, 3.77344807102863788791801830454, 3.90856056598605655124806739162, 4.33164030601427182734310720479, 4.41791254467034669397060970649, 4.45713504168616668148773390167, 4.66038776736180582045546907134, 4.71411978641204466280725551773, 5.09352510912561169219676630219, 5.27446975984209364591846858169, 5.45524109768114353801183822321, 5.61022646993261522194644557636, 5.73677781078729174526430216034, 5.95277900894735785534928954561, 5.98674601227753163190618742982

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