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Properties

Label 10-1076e5-1076.1075-c0e5-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.08864603903\)
\(L(\frac12)\) \(\approx\) \(0.08864603903\)
\(L(1)\) 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

−6.29500610572746255144434088923, −6.27196970130184967910212466651, −6.07783344392953565507987131862, −6.04367063974870964948178848141, −5.64112467350238841908376945880, −5.27224217193850584709673145495, −5.13282016908315859713536853542, −5.12851888296405194827810245824, −4.97437940880310271972820450145, −4.35357643616176874588211239033, −4.21490409369185583152354513373, −3.64647319375525524111396582421, −3.62213224484408251182335443223, −3.57269423229737068544080713095, −3.12719304120303015205993560836, −2.96069464342631444941296635265, −2.72478251761214302156259745783, −2.67360020806431388861067038398, −2.31772586807660821038445384757, −2.19592250178088381631973982664, −1.69574726302583715102049754556, −1.54381117407929401573209659719, −1.52082039940331860886146826744, −0.990338710799476993277972843449, −0.51567051495631759332122356752, 0.51567051495631759332122356752, 0.990338710799476993277972843449, 1.52082039940331860886146826744, 1.54381117407929401573209659719, 1.69574726302583715102049754556, 2.19592250178088381631973982664, 2.31772586807660821038445384757, 2.67360020806431388861067038398, 2.72478251761214302156259745783, 2.96069464342631444941296635265, 3.12719304120303015205993560836, 3.57269423229737068544080713095, 3.62213224484408251182335443223, 3.64647319375525524111396582421, 4.21490409369185583152354513373, 4.35357643616176874588211239033, 4.97437940880310271972820450145, 5.12851888296405194827810245824, 5.13282016908315859713536853542, 5.27224217193850584709673145495, 5.64112467350238841908376945880, 6.04367063974870964948178848141, 6.07783344392953565507987131862, 6.27196970130184967910212466651, 6.29500610572746255144434088923

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