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Properties

Label 10-6042e5-1.1-c1e5-0-0
Degree $10$
Conductor $8.052\times 10^{18}$
Sign $1$
Analytic cond. $2.61390\times 10^{8}$
Root an. cond. $6.94590$
Motivic weight $1$
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 − 5·3-s + 15·4-s + 2·5-s − 25·6-s + 2·7-s + 35·8-s + 15·9-s + 10·10-s − 3·11-s − 75·12-s + 7·13-s + 10·14-s − 10·15-s + 70·16-s + 11·17-s + 75·18-s + 5·19-s + 30·20-s − 10·21-s − 15·22-s + 4·23-s − 175·24-s − 12·25-s + 35·26-s − 35·27-s + 30·28-s + ⋯
L(s)  = 1  + 3.53·2-s − 2.88·3-s + 15/2·4-s + 0.894·5-s − 10.2·6-s + 0.755·7-s + 12.3·8-s + 5·9-s + 3.16·10-s − 0.904·11-s − 21.6·12-s + 1.94·13-s + 2.67·14-s − 2.58·15-s + 35/2·16-s + 2.66·17-s + 17.6·18-s + 1.14·19-s + 6.70·20-s − 2.18·21-s − 3.19·22-s + 0.834·23-s − 35.7·24-s − 2.39·25-s + 6.86·26-s − 6.73·27-s + 5.66·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 19^{5} \cdot 53^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 19^{5} \cdot 53^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{5} \cdot 3^{5} \cdot 19^{5} \cdot 53^{5}\)
Sign: $1$
Analytic conductor: \(2.61390\times 10^{8}\)
Root analytic conductor: \(6.94590\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{5} \cdot 3^{5} \cdot 19^{5} \cdot 53^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(128.9036240\)
\(L(\frac12)\) \(\approx\) \(128.9036240\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{5} \)
3$C_1$ \( ( 1 + T )^{5} \)
19$C_1$ \( ( 1 - T )^{5} \)
53$C_1$ \( ( 1 - T )^{5} \)
good5$C_2 \wr S_5$ \( 1 - 2 T + 16 T^{2} - 7 p T^{3} + 134 T^{4} - 244 T^{5} + 134 p T^{6} - 7 p^{3} T^{7} + 16 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 2 T + 16 T^{2} - 39 T^{3} + 26 p T^{4} - 330 T^{5} + 26 p^{2} T^{6} - 39 p^{2} T^{7} + 16 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 3 T + 31 T^{2} + 5 p T^{3} + 42 p T^{4} + 676 T^{5} + 42 p^{2} T^{6} + 5 p^{3} T^{7} + 31 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 7 T + 61 T^{2} - 279 T^{3} + 1416 T^{4} - 4848 T^{5} + 1416 p T^{6} - 279 p^{2} T^{7} + 61 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 11 T + 89 T^{2} - 509 T^{3} + 156 p T^{4} - 11484 T^{5} + 156 p^{2} T^{6} - 509 p^{2} T^{7} + 89 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 4 T + 38 T^{2} - 105 T^{3} - 292 T^{4} - 514 T^{5} - 292 p T^{6} - 105 p^{2} T^{7} + 38 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 5 T + 106 T^{2} - 470 T^{3} + 5456 T^{4} - 18596 T^{5} + 5456 p T^{6} - 470 p^{2} T^{7} + 106 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 5 T + 135 T^{2} - 16 p T^{3} + 7630 T^{4} - 21206 T^{5} + 7630 p T^{6} - 16 p^{3} T^{7} + 135 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 3 T + 161 T^{2} + 367 T^{3} + 11070 T^{4} + 19136 T^{5} + 11070 p T^{6} + 367 p^{2} T^{7} + 161 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 24 T + 400 T^{2} - 4519 T^{3} + 41023 T^{4} - 289010 T^{5} + 41023 p T^{6} - 4519 p^{2} T^{7} + 400 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 6 T + 126 T^{2} - 111 T^{3} + 4236 T^{4} + 354 p T^{5} + 4236 p T^{6} - 111 p^{2} T^{7} + 126 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 23 T + 378 T^{2} + 4421 T^{3} + 41517 T^{4} + 314152 T^{5} + 41517 p T^{6} + 4421 p^{2} T^{7} + 378 p^{3} T^{8} + 23 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 11 T + 270 T^{2} - 2240 T^{3} + 30618 T^{4} - 188132 T^{5} + 30618 p T^{6} - 2240 p^{2} T^{7} + 270 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 16 T + 288 T^{2} - 3261 T^{3} + 33585 T^{4} - 281770 T^{5} + 33585 p T^{6} - 3261 p^{2} T^{7} + 288 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 4 T + 246 T^{2} - 657 T^{3} + 27996 T^{4} - 56490 T^{5} + 27996 p T^{6} - 657 p^{2} T^{7} + 246 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 10 T + 191 T^{2} + 1624 T^{3} + 13254 T^{4} + 128892 T^{5} + 13254 p T^{6} + 1624 p^{2} T^{7} + 191 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 9 T + 251 T^{2} - 2005 T^{3} + 32202 T^{4} - 192248 T^{5} + 32202 p T^{6} - 2005 p^{2} T^{7} + 251 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 17 T + 415 T^{2} + 4563 T^{3} + 64494 T^{4} + 512600 T^{5} + 64494 p T^{6} + 4563 p^{2} T^{7} + 415 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 9 T + 329 T^{2} - 2539 T^{3} + 47834 T^{4} - 299752 T^{5} + 47834 p T^{6} - 2539 p^{2} T^{7} + 329 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 5 T + 380 T^{2} + 1486 T^{3} + 62174 T^{4} + 185418 T^{5} + 62174 p T^{6} + 1486 p^{2} T^{7} + 380 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 13 T + 435 T^{2} - 3685 T^{3} + 73316 T^{4} - 461868 T^{5} + 73316 p T^{6} - 3685 p^{2} T^{7} + 435 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.72026607816879875567095409968, −4.69283665791393413815695787252, −4.57590972137369574269242286158, −4.53633060812114394247022608315, −4.15689093009620450619558408563, −3.91357540097810975969297867332, −3.89249589612181169617534338430, −3.84886586778112100248835907466, −3.57505219049881961802231560265, −3.52120505463010978622734404378, −3.14302781710099511234061984553, −3.11046236056776607048949762373, −2.83784381233275323002547280165, −2.74580912821047955049152008363, −2.74368546760179541070416129557, −2.01511433395357136286741018331, −1.98813933151723854472824758539, −1.98185261877936894811218700321, −1.69693489282955314228736010469, −1.69166657121786398010653644705, −1.19600014047162125989784181049, −0.919912457353224874870658628106, −0.908642438942763403568206220041, −0.808257939110165218915163274356, −0.47067442985540866445043363595, 0.47067442985540866445043363595, 0.808257939110165218915163274356, 0.908642438942763403568206220041, 0.919912457353224874870658628106, 1.19600014047162125989784181049, 1.69166657121786398010653644705, 1.69693489282955314228736010469, 1.98185261877936894811218700321, 1.98813933151723854472824758539, 2.01511433395357136286741018331, 2.74368546760179541070416129557, 2.74580912821047955049152008363, 2.83784381233275323002547280165, 3.11046236056776607048949762373, 3.14302781710099511234061984553, 3.52120505463010978622734404378, 3.57505219049881961802231560265, 3.84886586778112100248835907466, 3.89249589612181169617534338430, 3.91357540097810975969297867332, 4.15689093009620450619558408563, 4.53633060812114394247022608315, 4.57590972137369574269242286158, 4.69283665791393413815695787252, 4.72026607816879875567095409968

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