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Properties

Label 10-4002e5-1.1-c1e5-0-0
Degree $10$
Conductor $1.027\times 10^{18}$
Sign $-1$
Analytic cond. $3.33250\times 10^{7}$
Root an. cond. $5.65297$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s − 5·3-s + 15·4-s + 5-s + 25·6-s − 4·7-s − 35·8-s + 15·9-s − 5·10-s + 5·11-s − 75·12-s − 11·13-s + 20·14-s − 5·15-s + 70·16-s + 4·17-s − 75·18-s − 4·19-s + 15·20-s + 20·21-s − 25·22-s − 5·23-s + 175·24-s − 6·25-s + 55·26-s − 35·27-s − 60·28-s + ⋯
L(s)  = 1  − 3.53·2-s − 2.88·3-s + 15/2·4-s + 0.447·5-s + 10.2·6-s − 1.51·7-s − 12.3·8-s + 5·9-s − 1.58·10-s + 1.50·11-s − 21.6·12-s − 3.05·13-s + 5.34·14-s − 1.29·15-s + 35/2·16-s + 0.970·17-s − 17.6·18-s − 0.917·19-s + 3.35·20-s + 4.36·21-s − 5.33·22-s − 1.04·23-s + 35.7·24-s − 6/5·25-s + 10.7·26-s − 6.73·27-s − 11.3·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 23^{5} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 23^{5} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{5} \cdot 3^{5} \cdot 23^{5} \cdot 29^{5}\)
Sign: $-1$
Analytic conductor: \(3.33250\times 10^{7}\)
Root analytic conductor: \(5.65297\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{5} \cdot 3^{5} \cdot 23^{5} \cdot 29^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{5} \)
23$C_1$ \( ( 1 + T )^{5} \)
29$C_1$ \( ( 1 + T )^{5} \)
good5$C_2 \wr S_5$ \( 1 - T + 7 T^{2} - 16 T^{3} + 13 p T^{4} - 61 T^{5} + 13 p^{2} T^{6} - 16 p^{2} T^{7} + 7 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 4 T + 4 p T^{2} + 86 T^{3} + 363 T^{4} + 832 T^{5} + 363 p T^{6} + 86 p^{2} T^{7} + 4 p^{4} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 5 T + 38 T^{2} - p^{2} T^{3} + 527 T^{4} - 1410 T^{5} + 527 p T^{6} - p^{4} T^{7} + 38 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 11 T + 102 T^{2} + 603 T^{3} + 3111 T^{4} + 11962 T^{5} + 3111 p T^{6} + 603 p^{2} T^{7} + 102 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 4 T + 30 T^{2} - 118 T^{3} + 753 T^{4} - 3464 T^{5} + 753 p T^{6} - 118 p^{2} T^{7} + 30 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 4 T + 40 T^{2} + 150 T^{3} + 1143 T^{4} + 4576 T^{5} + 1143 p T^{6} + 150 p^{2} T^{7} + 40 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 5 T + 68 T^{2} + 11 T^{3} + 697 T^{4} + 10018 T^{5} + 697 p T^{6} + 11 p^{2} T^{7} + 68 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 9 T + 119 T^{2} - 1024 T^{3} + 8109 T^{4} - 1337 p T^{5} + 8109 p T^{6} - 1024 p^{2} T^{7} + 119 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 5 T + 169 T^{2} - 608 T^{3} + 12377 T^{4} - 33861 T^{5} + 12377 p T^{6} - 608 p^{2} T^{7} + 169 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 16 T + 280 T^{2} + 2730 T^{3} + 26615 T^{4} + 175444 T^{5} + 26615 p T^{6} + 2730 p^{2} T^{7} + 280 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 8 T + 48 T^{2} - 274 T^{3} + 1907 T^{4} - 25496 T^{5} + 1907 p T^{6} - 274 p^{2} T^{7} + 48 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 4 T + 109 T^{2} + 532 T^{3} + 6806 T^{4} + 29440 T^{5} + 6806 p T^{6} + 532 p^{2} T^{7} + 109 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 23 T + 421 T^{2} - 5154 T^{3} + 55025 T^{4} - 451553 T^{5} + 55025 p T^{6} - 5154 p^{2} T^{7} + 421 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 7 T + 54 T^{2} - 1097 T^{3} + 8949 T^{4} - 35032 T^{5} + 8949 p T^{6} - 1097 p^{2} T^{7} + 54 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 5 T + 178 T^{2} - 381 T^{3} + 16941 T^{4} - 29612 T^{5} + 16941 p T^{6} - 381 p^{2} T^{7} + 178 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 21 T + 452 T^{2} + 5785 T^{3} + 69929 T^{4} + 608650 T^{5} + 69929 p T^{6} + 5785 p^{2} T^{7} + 452 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 8 T + 49 T^{2} + 28 T^{3} + 7398 T^{4} + 71192 T^{5} + 7398 p T^{6} + 28 p^{2} T^{7} + 49 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 8 T + 393 T^{2} + 2460 T^{3} + 62016 T^{4} + 288800 T^{5} + 62016 p T^{6} + 2460 p^{2} T^{7} + 393 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 18 T + 103 T^{2} + 168 T^{3} + 134 p T^{4} + 192460 T^{5} + 134 p^{2} T^{6} + 168 p^{2} T^{7} + 103 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 32 T + 665 T^{2} - 9820 T^{3} + 122750 T^{4} - 1249656 T^{5} + 122750 p T^{6} - 9820 p^{2} T^{7} + 665 p^{3} T^{8} - 32 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 16 T + 425 T^{2} + 4908 T^{3} + 75446 T^{4} + 664696 T^{5} + 75446 p T^{6} + 4908 p^{2} T^{7} + 425 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.47262664438804258936015800793, −5.29863057174063296530974905829, −5.21957185567357356586545133267, −5.08994522946566793733403671114, −5.07698801855859205716715204747, −4.53573558772942794330289448696, −4.39927174011558459933891701962, −4.37330929900252047456446448045, −4.09950670220688562053097865653, −4.04358760341228978143757835957, −3.60961268161993976456194889568, −3.55835338934226451363465026818, −3.41441025034132854064354956597, −3.06714987111835461054128383817, −2.97275004730072728481209975362, −2.46244279937793055891075075847, −2.44988023498937485379709386146, −2.21989771787079044870588828463, −2.21055314502806863532332700126, −2.14371433237439539368538134572, −1.44490116772044788147592720331, −1.30059204841024934312259414388, −1.27576154323326769908535480658, −1.15985899128314676835907903095, −0.937170139549211323205395888021, 0, 0, 0, 0, 0, 0.937170139549211323205395888021, 1.15985899128314676835907903095, 1.27576154323326769908535480658, 1.30059204841024934312259414388, 1.44490116772044788147592720331, 2.14371433237439539368538134572, 2.21055314502806863532332700126, 2.21989771787079044870588828463, 2.44988023498937485379709386146, 2.46244279937793055891075075847, 2.97275004730072728481209975362, 3.06714987111835461054128383817, 3.41441025034132854064354956597, 3.55835338934226451363465026818, 3.60961268161993976456194889568, 4.04358760341228978143757835957, 4.09950670220688562053097865653, 4.37330929900252047456446448045, 4.39927174011558459933891701962, 4.53573558772942794330289448696, 5.07698801855859205716715204747, 5.08994522946566793733403671114, 5.21957185567357356586545133267, 5.29863057174063296530974905829, 5.47262664438804258936015800793

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