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Properties

Label 10-8512e5-1.1-c1e5-0-0
Degree $10$
Conductor $4.468\times 10^{19}$
Sign $1$
Analytic cond. $1.45058\times 10^{9}$
Root an. cond. $8.24431$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 5·7-s − 4·9-s + 6·11-s − 4·13-s + 9·17-s + 5·19-s + 6·23-s − 2·25-s − 10·29-s + 31-s − 15·35-s − 17·37-s + 24·41-s + 15·43-s + 12·45-s − 5·47-s + 15·49-s − 8·53-s − 18·55-s − 5·59-s − 9·61-s − 20·63-s + 12·65-s + 17·67-s + 71-s − 17·73-s + ⋯
L(s)  = 1  − 1.34·5-s + 1.88·7-s − 4/3·9-s + 1.80·11-s − 1.10·13-s + 2.18·17-s + 1.14·19-s + 1.25·23-s − 2/5·25-s − 1.85·29-s + 0.179·31-s − 2.53·35-s − 2.79·37-s + 3.74·41-s + 2.28·43-s + 1.78·45-s − 0.729·47-s + 15/7·49-s − 1.09·53-s − 2.42·55-s − 0.650·59-s − 1.15·61-s − 2.51·63-s + 1.48·65-s + 2.07·67-s + 0.118·71-s − 1.98·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{30} \cdot 7^{5} \cdot 19^{5}\)
Sign: $1$
Analytic conductor: \(1.45058\times 10^{9}\)
Root analytic conductor: \(8.24431\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{30} \cdot 7^{5} \cdot 19^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.758093985\)
\(L(\frac12)\) \(\approx\) \(4.758093985\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{5} \)
19$C_1$ \( ( 1 - T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + 4 T^{2} + 8 T^{4} + 4 T^{5} + 8 p T^{6} + 4 p^{3} T^{8} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 3 T + 11 T^{2} + 21 T^{3} + 51 T^{4} + 82 T^{5} + 51 p T^{6} + 21 p^{2} T^{7} + 11 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 6 T + 36 T^{2} - 168 T^{3} + 58 p T^{4} - 2280 T^{5} + 58 p^{2} T^{6} - 168 p^{2} T^{7} + 36 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 4 T + 40 T^{2} + 120 T^{3} + 779 T^{4} + 1760 T^{5} + 779 p T^{6} + 120 p^{2} T^{7} + 40 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 9 T + 80 T^{2} - 438 T^{3} + 2427 T^{4} - 9914 T^{5} + 2427 p T^{6} - 438 p^{2} T^{7} + 80 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 6 T + 98 T^{2} - 474 T^{3} + 4185 T^{4} - 15712 T^{5} + 4185 p T^{6} - 474 p^{2} T^{7} + 98 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 10 T + 94 T^{2} + 766 T^{3} + 5338 T^{4} + 28054 T^{5} + 5338 p T^{6} + 766 p^{2} T^{7} + 94 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - T + 118 T^{2} - 132 T^{3} + 6389 T^{4} - 6166 T^{5} + 6389 p T^{6} - 132 p^{2} T^{7} + 118 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 17 T + 241 T^{2} + 2289 T^{3} + 19241 T^{4} + 123814 T^{5} + 19241 p T^{6} + 2289 p^{2} T^{7} + 241 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 24 T + 318 T^{2} - 2674 T^{3} + 17488 T^{4} - 105894 T^{5} + 17488 p T^{6} - 2674 p^{2} T^{7} + 318 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 15 T + 160 T^{2} - 740 T^{3} + 1999 T^{4} + 5926 T^{5} + 1999 p T^{6} - 740 p^{2} T^{7} + 160 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 5 T + 159 T^{2} + 959 T^{3} + 12245 T^{4} + 66832 T^{5} + 12245 p T^{6} + 959 p^{2} T^{7} + 159 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 8 T + 150 T^{2} + 1238 T^{3} + 13352 T^{4} + 85918 T^{5} + 13352 p T^{6} + 1238 p^{2} T^{7} + 150 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 5 T + 113 T^{2} + 335 T^{3} + 9801 T^{4} + 33428 T^{5} + 9801 p T^{6} + 335 p^{2} T^{7} + 113 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 9 T + 231 T^{2} + 1599 T^{3} + 24939 T^{4} + 137618 T^{5} + 24939 p T^{6} + 1599 p^{2} T^{7} + 231 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 17 T + 256 T^{2} - 2772 T^{3} + 25751 T^{4} - 232294 T^{5} + 25751 p T^{6} - 2772 p^{2} T^{7} + 256 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - T + 93 T^{2} - 727 T^{3} + 10949 T^{4} - 35796 T^{5} + 10949 p T^{6} - 727 p^{2} T^{7} + 93 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 17 T + 112 T^{2} + 408 T^{3} + 13739 T^{4} + 184246 T^{5} + 13739 p T^{6} + 408 p^{2} T^{7} + 112 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 13 T + 332 T^{2} + 3472 T^{3} + 49963 T^{4} + 384438 T^{5} + 49963 p T^{6} + 3472 p^{2} T^{7} + 332 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 15 T + 408 T^{2} + 4400 T^{3} + 66283 T^{4} + 523986 T^{5} + 66283 p T^{6} + 4400 p^{2} T^{7} + 408 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 29 T + 582 T^{2} - 8648 T^{3} + 108269 T^{4} - 1119502 T^{5} + 108269 p T^{6} - 8648 p^{2} T^{7} + 582 p^{3} T^{8} - 29 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 7 T + 247 T^{2} + 435 T^{3} + 20045 T^{4} - 45242 T^{5} + 20045 p T^{6} + 435 p^{2} T^{7} + 247 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.52591103833182670618239462977, −4.30567385413889005135027965185, −4.25059027924289658308396197828, −4.06157766478372353386467850614, −4.01663785142111012130732785132, −3.90855231637600890440249631271, −3.58173890620408037151391839174, −3.43778836014144860029015875225, −3.39586244400274293932644433884, −3.30275549765263914387153666733, −2.93383994663995730005767187454, −2.89771189915173703677514257787, −2.65968058030687902308428685034, −2.49142100063390392586545600037, −2.35182225029130728172585895595, −2.25687222282110423147289899942, −1.78425095713791648722439509752, −1.62918856753308146876955983756, −1.48335834785612485841788020559, −1.37662547475130153548395775502, −1.23923885270082469830402833265, −1.00422390008245555333562346675, −0.64189022671967826016507949016, −0.44663407953367541758812240245, −0.23310882608755625575716802499, 0.23310882608755625575716802499, 0.44663407953367541758812240245, 0.64189022671967826016507949016, 1.00422390008245555333562346675, 1.23923885270082469830402833265, 1.37662547475130153548395775502, 1.48335834785612485841788020559, 1.62918856753308146876955983756, 1.78425095713791648722439509752, 2.25687222282110423147289899942, 2.35182225029130728172585895595, 2.49142100063390392586545600037, 2.65968058030687902308428685034, 2.89771189915173703677514257787, 2.93383994663995730005767187454, 3.30275549765263914387153666733, 3.39586244400274293932644433884, 3.43778836014144860029015875225, 3.58173890620408037151391839174, 3.90855231637600890440249631271, 4.01663785142111012130732785132, 4.06157766478372353386467850614, 4.25059027924289658308396197828, 4.30567385413889005135027965185, 4.52591103833182670618239462977

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