↑

Properties

Label 10-4508e5-1.1-c1e5-0-1
Degree $10$
Conductor $1.862\times 10^{18}$
Sign $-1$
Analytic cond. $6.04373\times 10^{7}$
Root an. cond. $5.99971$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 2·5-s + 2·9-s + 2·11-s − 13·13-s + 6·15-s − 4·17-s − 12·19-s − 5·23-s − 25-s + 27-s + 13·29-s + 3·31-s − 6·33-s − 4·37-s + 39·39-s − 41-s − 8·43-s − 4·45-s − 5·47-s + 12·51-s − 8·53-s − 4·55-s + 36·57-s − 12·59-s − 20·61-s + 26·65-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.894·5-s + 2/3·9-s + 0.603·11-s − 3.60·13-s + 1.54·15-s − 0.970·17-s − 2.75·19-s − 1.04·23-s − 1/5·25-s + 0.192·27-s + 2.41·29-s + 0.538·31-s − 1.04·33-s − 0.657·37-s + 6.24·39-s − 0.156·41-s − 1.21·43-s − 0.596·45-s − 0.729·47-s + 1.68·51-s − 1.09·53-s − 0.539·55-s + 4.76·57-s − 1.56·59-s − 2.56·61-s + 3.22·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 7^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 7^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{10} \cdot 7^{10} \cdot 23^{5}\)
Sign: $-1$
Analytic conductor: \(6.04373\times 10^{7}\)
Root analytic conductor: \(5.99971\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{10} \cdot 7^{10} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
7 \( 1 \)
23$C_1$ \( ( 1 + T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + p T + 7 T^{2} + 14 T^{3} + 34 T^{4} + 68 T^{5} + 34 p T^{6} + 14 p^{2} T^{7} + 7 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 2 T + p T^{2} + 6 T^{3} + 44 T^{4} + 24 p T^{5} + 44 p T^{6} + 6 p^{2} T^{7} + p^{4} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 2 T + 7 T^{2} - 20 T^{3} + 190 T^{4} - 316 T^{5} + 190 p T^{6} - 20 p^{2} T^{7} + 7 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + p T + 85 T^{2} + 356 T^{3} + 1110 T^{4} + 3438 T^{5} + 1110 p T^{6} + 356 p^{2} T^{7} + 85 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 4 T + 37 T^{2} + 202 T^{3} + 936 T^{4} + 4300 T^{5} + 936 p T^{6} + 202 p^{2} T^{7} + 37 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 12 T + 5 p T^{2} + 592 T^{3} + 3162 T^{4} + 14856 T^{5} + 3162 p T^{6} + 592 p^{2} T^{7} + 5 p^{4} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 13 T + 5 p T^{2} - 1204 T^{3} + 8338 T^{4} - 48762 T^{5} + 8338 p T^{6} - 1204 p^{2} T^{7} + 5 p^{4} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 3 T + 13 T^{2} + 198 T^{3} + 916 T^{4} - 2768 T^{5} + 916 p T^{6} + 198 p^{2} T^{7} + 13 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 4 T + 97 T^{2} + 80 T^{3} + 3682 T^{4} - 3880 T^{5} + 3682 p T^{6} + 80 p^{2} T^{7} + 97 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + T + 121 T^{2} + 68 T^{3} + 8206 T^{4} + 4246 T^{5} + 8206 p T^{6} + 68 p^{2} T^{7} + 121 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 8 T + 127 T^{2} + 352 T^{3} + 4338 T^{4} - 1296 T^{5} + 4338 p T^{6} + 352 p^{2} T^{7} + 127 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 5 T + 105 T^{2} + 298 T^{3} + 5648 T^{4} + 7120 T^{5} + 5648 p T^{6} + 298 p^{2} T^{7} + 105 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 8 T + 121 T^{2} + 752 T^{3} + 10234 T^{4} + 62992 T^{5} + 10234 p T^{6} + 752 p^{2} T^{7} + 121 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 12 T + 251 T^{2} + 2754 T^{3} + 27184 T^{4} + 241644 T^{5} + 27184 p T^{6} + 2754 p^{2} T^{7} + 251 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 20 T + 365 T^{2} + 4638 T^{3} + 47608 T^{4} + 416704 T^{5} + 47608 p T^{6} + 4638 p^{2} T^{7} + 365 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 12 T + 239 T^{2} + 1332 T^{3} + 17606 T^{4} + 61744 T^{5} + 17606 p T^{6} + 1332 p^{2} T^{7} + 239 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 9 T + 175 T^{2} - 332 T^{3} + 8806 T^{4} + 23754 T^{5} + 8806 p T^{6} - 332 p^{2} T^{7} + 175 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 9 T + 253 T^{2} - 2140 T^{3} + 32218 T^{4} - 217814 T^{5} + 32218 p T^{6} - 2140 p^{2} T^{7} + 253 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 8 T + 235 T^{2} + 620 T^{3} + 18278 T^{4} - 104 p T^{5} + 18278 p T^{6} + 620 p^{2} T^{7} + 235 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 28 T + 575 T^{2} - 8448 T^{3} + 102170 T^{4} - 1012488 T^{5} + 102170 p T^{6} - 8448 p^{2} T^{7} + 575 p^{3} T^{8} - 28 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 32 T + 593 T^{2} + 8038 T^{3} + 91908 T^{4} + 923620 T^{5} + 91908 p T^{6} + 8038 p^{2} T^{7} + 593 p^{3} T^{8} + 32 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 4 T + 289 T^{2} + 1438 T^{3} + 47792 T^{4} + 176000 T^{5} + 47792 p T^{6} + 1438 p^{2} T^{7} + 289 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.18428109915608002401264877204, −5.01374274742327470630517853902, −4.88140894198853507059080503597, −4.85784261994273133570909314523, −4.76376634125184019382164429418, −4.61382085190785092706353395744, −4.54519994102965703095618184947, −4.29522322532694027991924597175, −4.05401587845745428322267920036, −4.03624831561775571646091411115, −3.80700011708833684171317244059, −3.56304776088005295237262723774, −3.34971818437366499074867971210, −3.28364755380002382521052045272, −2.80301739798034986301624335936, −2.65352543982543788689524183005, −2.61902624403840900958483747556, −2.54988950029611069243085268655, −2.38056940886870610253762671777, −1.90668629279762800598438997206, −1.90207984410425197853713466769, −1.59918829242567361610174694541, −1.41951682733602853487635138474, −1.11751148130416146385011373280, −0.797366571711942069546930756780, 0, 0, 0, 0, 0, 0.797366571711942069546930756780, 1.11751148130416146385011373280, 1.41951682733602853487635138474, 1.59918829242567361610174694541, 1.90207984410425197853713466769, 1.90668629279762800598438997206, 2.38056940886870610253762671777, 2.54988950029611069243085268655, 2.61902624403840900958483747556, 2.65352543982543788689524183005, 2.80301739798034986301624335936, 3.28364755380002382521052045272, 3.34971818437366499074867971210, 3.56304776088005295237262723774, 3.80700011708833684171317244059, 4.03624831561775571646091411115, 4.05401587845745428322267920036, 4.29522322532694027991924597175, 4.54519994102965703095618184947, 4.61382085190785092706353395744, 4.76376634125184019382164429418, 4.85784261994273133570909314523, 4.88140894198853507059080503597, 5.01374274742327470630517853902, 5.18428109915608002401264877204

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