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Properties

Label 10-215e5-1.1-c1e5-0-0
Degree $10$
Conductor $459401384375$
Sign $1$
Analytic cond. $14.9134$
Root an. cond. $1.31026$
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  + 2·2-s − 3-s + 4-s + 5·5-s − 2·6-s + 5·7-s − 8-s + 2·9-s + 10·10-s − 6·11-s − 12-s + 5·13-s + 10·14-s − 5·15-s − 2·16-s − 17·17-s + 4·18-s − 6·19-s + 5·20-s − 5·21-s − 12·22-s + 23-s + 24-s + 15·25-s + 10·26-s − 8·27-s + 5·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 1/2·4-s + 2.23·5-s − 0.816·6-s + 1.88·7-s − 0.353·8-s + 2/3·9-s + 3.16·10-s − 1.80·11-s − 0.288·12-s + 1.38·13-s + 2.67·14-s − 1.29·15-s − 1/2·16-s − 4.12·17-s + 0.942·18-s − 1.37·19-s + 1.11·20-s − 1.09·21-s − 2.55·22-s + 0.208·23-s + 0.204·24-s + 3·25-s + 1.96·26-s − 1.53·27-s + 0.944·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 43^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 43^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(5^{5} \cdot 43^{5}\)
Sign: $1$
Analytic conductor: \(14.9134\)
Root analytic conductor: \(1.31026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 5^{5} \cdot 43^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.410518727\)
\(L(\frac12)\) \(\approx\) \(4.410518727\)
\(L(\frac{3}{2})\) 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)$
bad5$C_1$ \( ( 1 - T )^{5} \)
43$C_1$ \( ( 1 + T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - p T + 3 T^{2} - 3 T^{3} + 3 T^{4} + 3 p T^{6} - 3 p^{2} T^{7} + 3 p^{3} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 + T - T^{2} + 5 T^{3} + 10 T^{4} - 4 T^{5} + 10 p T^{6} + 5 p^{2} T^{7} - p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 5 T + 3 p T^{2} - 43 T^{3} + 138 T^{4} - 272 T^{5} + 138 p T^{6} - 43 p^{2} T^{7} + 3 p^{4} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 6 T + 56 T^{2} + 221 T^{3} + 1184 T^{4} + 3398 T^{5} + 1184 p T^{6} + 221 p^{2} T^{7} + 56 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 5 T + 15 T^{2} + 24 T^{3} - 36 T^{4} + 314 T^{5} - 36 p T^{6} + 24 p^{2} T^{7} + 15 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + p T + 179 T^{2} + 1336 T^{3} + 7764 T^{4} + 35582 T^{5} + 7764 p T^{6} + 1336 p^{2} T^{7} + 179 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 6 T + 23 T^{2} + 104 T^{3} + 786 T^{4} + 4228 T^{5} + 786 p T^{6} + 104 p^{2} T^{7} + 23 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - T + 61 T^{2} + 40 T^{3} + 1764 T^{4} + 2514 T^{5} + 1764 p T^{6} + 40 p^{2} T^{7} + 61 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 6 T + 61 T^{2} + 56 T^{3} - 642 T^{4} + 14492 T^{5} - 642 p T^{6} + 56 p^{2} T^{7} + 61 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 6 T + 88 T^{2} - 215 T^{3} + 2476 T^{4} - 1670 T^{5} + 2476 p T^{6} - 215 p^{2} T^{7} + 88 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 5 T + 157 T^{2} - 613 T^{3} + 10668 T^{4} - 32072 T^{5} + 10668 p T^{6} - 613 p^{2} T^{7} + 157 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 2 T + 106 T^{2} - 81 T^{3} + 4844 T^{4} + 112 T^{5} + 4844 p T^{6} - 81 p^{2} T^{7} + 106 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 111 T^{2} + 72 T^{3} + 7998 T^{4} + 4720 T^{5} + 7998 p T^{6} + 72 p^{2} T^{7} + 111 p^{3} T^{8} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 23 T + 455 T^{2} + 5544 T^{3} + 59212 T^{4} + 458850 T^{5} + 59212 p T^{6} + 5544 p^{2} T^{7} + 455 p^{3} T^{8} + 23 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + T + 279 T^{2} + 229 T^{3} + 32042 T^{4} + 20044 T^{5} + 32042 p T^{6} + 229 p^{2} T^{7} + 279 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 20 T + 225 T^{2} - 1728 T^{3} + 15066 T^{4} - 122648 T^{5} + 15066 p T^{6} - 1728 p^{2} T^{7} + 225 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 21 T + 379 T^{2} - 4896 T^{3} + 54318 T^{4} - 467430 T^{5} + 54318 p T^{6} - 4896 p^{2} T^{7} + 379 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 4 T + 143 T^{2} - 504 T^{3} + 15830 T^{4} - 51592 T^{5} + 15830 p T^{6} - 504 p^{2} T^{7} + 143 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 5 T + 281 T^{2} - 1269 T^{3} + 36116 T^{4} - 130872 T^{5} + 36116 p T^{6} - 1269 p^{2} T^{7} + 281 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 41 T + 1039 T^{2} - 17721 T^{3} + 231158 T^{4} - 2306844 T^{5} + 231158 p T^{6} - 17721 p^{2} T^{7} + 1039 p^{3} T^{8} - 41 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 7 T + 317 T^{2} + 1436 T^{3} + 43232 T^{4} + 144330 T^{5} + 43232 p T^{6} + 1436 p^{2} T^{7} + 317 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 20 T + 453 T^{2} - 6120 T^{3} + 80658 T^{4} - 775176 T^{5} + 80658 p T^{6} - 6120 p^{2} T^{7} + 453 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 37 T + 895 T^{2} - 15564 T^{3} + 213240 T^{4} - 2321998 T^{5} + 213240 p T^{6} - 15564 p^{2} T^{7} + 895 p^{3} T^{8} - 37 p^{4} T^{9} + p^{5} T^{10} \)
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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

−7.73215881408839433198756800413, −7.71504626870102717585930297892, −7.05647023996048391779964774293, −6.72666855071818776098386474037, −6.66339697382034996122215499561, −6.44931301245991756591961945737, −6.38660787669358764362633748220, −5.95584398233890488998745922872, −5.87514393191210427976811024877, −5.85756360357063328439501080491, −5.10096838705268795809203608294, −5.00124895182054605591664141680, −4.79916637245513282379813234948, −4.76687290163249739983128946101, −4.75935199864711992867525577511, −4.46618549365257646965858516962, −3.86267461588158781407795964967, −3.58069433572589727444331876071, −3.54827251858898717835910881923, −2.59830012752231205992161817386, −2.55109570930900472139917024915, −2.05451283986072788789024163054, −1.99492378410569780374581339054, −1.87397009868411036295435173279, −0.859673742817930622763433774650, 0.859673742817930622763433774650, 1.87397009868411036295435173279, 1.99492378410569780374581339054, 2.05451283986072788789024163054, 2.55109570930900472139917024915, 2.59830012752231205992161817386, 3.54827251858898717835910881923, 3.58069433572589727444331876071, 3.86267461588158781407795964967, 4.46618549365257646965858516962, 4.75935199864711992867525577511, 4.76687290163249739983128946101, 4.79916637245513282379813234948, 5.00124895182054605591664141680, 5.10096838705268795809203608294, 5.85756360357063328439501080491, 5.87514393191210427976811024877, 5.95584398233890488998745922872, 6.38660787669358764362633748220, 6.44931301245991756591961945737, 6.66339697382034996122215499561, 6.72666855071818776098386474037, 7.05647023996048391779964774293, 7.71504626870102717585930297892, 7.73215881408839433198756800413

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