Properties

Label 10-69e5-1.1-c5e5-0-0
Degree $10$
Conductor $1564031349$
Sign $1$
Analytic cond. $165977.$
Root an. cond. $3.32663$
Motivic weight $5$
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  + 8·2-s + 45·3-s + 11·4-s + 94·5-s + 360·6-s + 272·7-s − 82·8-s + 1.21e3·9-s + 752·10-s + 1.10e3·11-s + 495·12-s − 978·13-s + 2.17e3·14-s + 4.23e3·15-s − 915·16-s + 2.52e3·17-s + 9.72e3·18-s + 2.06e3·19-s + 1.03e3·20-s + 1.22e4·21-s + 8.80e3·22-s − 2.64e3·23-s − 3.69e3·24-s + 2.62e3·25-s − 7.82e3·26-s + 2.55e4·27-s + 2.99e3·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.88·3-s + 0.343·4-s + 1.68·5-s + 4.08·6-s + 2.09·7-s − 0.452·8-s + 5·9-s + 2.37·10-s + 2.74·11-s + 0.992·12-s − 1.60·13-s + 2.96·14-s + 4.85·15-s − 0.893·16-s + 2.11·17-s + 7.07·18-s + 1.30·19-s + 0.578·20-s + 6.05·21-s + 3.87·22-s − 1.04·23-s − 1.30·24-s + 0.839·25-s − 2.26·26-s + 6.73·27-s + 0.721·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{5} \cdot 23^{5}\)
Sign: $1$
Analytic conductor: \(165977.\)
Root analytic conductor: \(3.32663\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{69} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 3^{5} \cdot 23^{5} ,\ ( \ : 5/2, 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(103.7933272\)
\(L(\frac12)\) \(\approx\) \(103.7933272\)
\(L(\frac{7}{2})\) 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)$
bad3$C_1$ \( ( 1 - p^{2} T )^{5} \)
23$C_1$ \( ( 1 + p^{2} T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - p^{3} T + 53 T^{2} - 127 p T^{3} + 427 p^{2} T^{4} - 789 p^{3} T^{5} + 427 p^{7} T^{6} - 127 p^{11} T^{7} + 53 p^{15} T^{8} - p^{23} T^{9} + p^{25} T^{10} \)
5$C_2 \wr S_5$ \( 1 - 94 T + 6213 T^{2} - 233272 T^{3} + 16438526 T^{4} - 918832724 T^{5} + 16438526 p^{5} T^{6} - 233272 p^{10} T^{7} + 6213 p^{15} T^{8} - 94 p^{20} T^{9} + p^{25} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 272 T + 92999 T^{2} - 15921576 T^{3} + 3139766110 T^{4} - 382935152112 T^{5} + 3139766110 p^{5} T^{6} - 15921576 p^{10} T^{7} + 92999 p^{15} T^{8} - 272 p^{20} T^{9} + p^{25} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 100 p T + 950167 T^{2} - 47832944 p T^{3} + 266170957962 T^{4} - 106836876092680 T^{5} + 266170957962 p^{5} T^{6} - 47832944 p^{11} T^{7} + 950167 p^{15} T^{8} - 100 p^{21} T^{9} + p^{25} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 978 T + 1491369 T^{2} + 962007576 T^{3} + 67101754522 p T^{4} + 442247737686732 T^{5} + 67101754522 p^{6} T^{6} + 962007576 p^{10} T^{7} + 1491369 p^{15} T^{8} + 978 p^{20} T^{9} + p^{25} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 2522 T + 8504033 T^{2} - 13985869736 T^{3} + 25829300837998 T^{4} - 29533869359617308 T^{5} + 25829300837998 p^{5} T^{6} - 13985869736 p^{10} T^{7} + 8504033 p^{15} T^{8} - 2522 p^{20} T^{9} + p^{25} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 2060 T + 429073 p T^{2} - 6053371160 T^{3} + 17747857284814 T^{4} - 1707014975296600 T^{5} + 17747857284814 p^{5} T^{6} - 6053371160 p^{10} T^{7} + 429073 p^{16} T^{8} - 2060 p^{20} T^{9} + p^{25} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 1526 T + 53450201 T^{2} - 160216816776 T^{3} + 1463826216452722 T^{4} - 4967385842939808356 T^{5} + 1463826216452722 p^{5} T^{6} - 160216816776 p^{10} T^{7} + 53450201 p^{15} T^{8} - 1526 p^{20} T^{9} + p^{25} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 7392 T + 143582859 T^{2} + 814586108224 T^{3} + 8222478851768122 T^{4} + 34540989938511230912 T^{5} + 8222478851768122 p^{5} T^{6} + 814586108224 p^{10} T^{7} + 143582859 p^{15} T^{8} + 7392 p^{20} T^{9} + p^{25} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 8210 T + 333147057 T^{2} + 2158433364600 T^{3} + 45276559134316162 T^{4} + \)\(22\!\cdots\!00\)\( T^{5} + 45276559134316162 p^{5} T^{6} + 2158433364600 p^{10} T^{7} + 333147057 p^{15} T^{8} + 8210 p^{20} T^{9} + p^{25} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 21250 T + 285983317 T^{2} - 1967560460248 T^{3} + 17015437600068850 T^{4} - \)\(11\!\cdots\!08\)\( T^{5} + 17015437600068850 p^{5} T^{6} - 1967560460248 p^{10} T^{7} + 285983317 p^{15} T^{8} - 21250 p^{20} T^{9} + p^{25} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 4548 T + 287763707 T^{2} + 3207650046632 T^{3} + 68028185295500398 T^{4} + \)\(50\!\cdots\!76\)\( T^{5} + 68028185295500398 p^{5} T^{6} + 3207650046632 p^{10} T^{7} + 287763707 p^{15} T^{8} + 4548 p^{20} T^{9} + p^{25} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 536 T + 264020283 T^{2} - 2501526488736 T^{3} + 112580424453867674 T^{4} - \)\(14\!\cdots\!12\)\( T^{5} + 112580424453867674 p^{5} T^{6} - 2501526488736 p^{10} T^{7} + 264020283 p^{15} T^{8} - 536 p^{20} T^{9} + p^{25} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 11482 T + 1116302917 T^{2} - 941657798856 T^{3} + 483029536217021550 T^{4} - \)\(40\!\cdots\!16\)\( T^{5} + 483029536217021550 p^{5} T^{6} - 941657798856 p^{10} T^{7} + 1116302917 p^{15} T^{8} + 11482 p^{20} T^{9} + p^{25} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 74676 T + 4089190183 T^{2} - 163708505958576 T^{3} + 5812674984278443178 T^{4} - \)\(16\!\cdots\!84\)\( T^{5} + 5812674984278443178 p^{5} T^{6} - 163708505958576 p^{10} T^{7} + 4089190183 p^{15} T^{8} - 74676 p^{20} T^{9} + p^{25} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 44618 T + 4437282569 T^{2} + 138965692205272 T^{3} + 7551174855833493282 T^{4} + \)\(17\!\cdots\!68\)\( T^{5} + 7551174855833493282 p^{5} T^{6} + 138965692205272 p^{10} T^{7} + 4437282569 p^{15} T^{8} + 44618 p^{20} T^{9} + p^{25} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 1412 T + 4040959155 T^{2} - 20057652231432 T^{3} + 7903786473880913422 T^{4} - \)\(60\!\cdots\!48\)\( T^{5} + 7903786473880913422 p^{5} T^{6} - 20057652231432 p^{10} T^{7} + 4040959155 p^{15} T^{8} + 1412 p^{20} T^{9} + p^{25} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 37912 T + 1466467779 T^{2} + 7879531293536 T^{3} + 2981493544848827562 T^{4} - \)\(14\!\cdots\!32\)\( T^{5} + 2981493544848827562 p^{5} T^{6} + 7879531293536 p^{10} T^{7} + 1466467779 p^{15} T^{8} - 37912 p^{20} T^{9} + p^{25} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 46546 T + 8346436661 T^{2} - 319480387049752 T^{3} + 31398063765510760370 T^{4} - \)\(92\!\cdots\!96\)\( T^{5} + 31398063765510760370 p^{5} T^{6} - 319480387049752 p^{10} T^{7} + 8346436661 p^{15} T^{8} - 46546 p^{20} T^{9} + p^{25} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 50544 T + 6561571119 T^{2} - 53943736845944 T^{3} + 15591070525869009022 T^{4} + \)\(22\!\cdots\!12\)\( T^{5} + 15591070525869009022 p^{5} T^{6} - 53943736845944 p^{10} T^{7} + 6561571119 p^{15} T^{8} - 50544 p^{20} T^{9} + p^{25} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 89588 T + 10451924351 T^{2} - 775319273971120 T^{3} + 61260146630661454026 T^{4} - \)\(36\!\cdots\!20\)\( T^{5} + 61260146630661454026 p^{5} T^{6} - 775319273971120 p^{10} T^{7} + 10451924351 p^{15} T^{8} - 89588 p^{20} T^{9} + p^{25} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 280410 T + 47711380473 T^{2} - 5653018849483864 T^{3} + \)\(53\!\cdots\!70\)\( T^{4} - \)\(42\!\cdots\!76\)\( T^{5} + \)\(53\!\cdots\!70\)\( p^{5} T^{6} - 5653018849483864 p^{10} T^{7} + 47711380473 p^{15} T^{8} - 280410 p^{20} T^{9} + p^{25} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 90074 T + 36682650013 T^{2} - 2072067856619224 T^{3} + \)\(53\!\cdots\!94\)\( T^{4} - \)\(21\!\cdots\!28\)\( T^{5} + \)\(53\!\cdots\!94\)\( p^{5} T^{6} - 2072067856619224 p^{10} T^{7} + 36682650013 p^{15} T^{8} - 90074 p^{20} T^{9} + p^{25} T^{10} \)
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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

−8.165266536736616001604929017023, −8.131878632539373894394789052338, −8.037280753225211284468601151860, −7.61793210200202970135012910545, −7.41716477950073777319290556333, −7.12167511553267171037756707280, −7.04942233489018741394881458200, −6.25115543635079235497105364240, −6.24162901151056270447519969965, −6.02342953779757556066240162203, −5.35898668893587231901003368819, −5.11230797549031381038360408317, −4.94715596843835069023723653343, −4.57896778918635400770656627339, −4.54929984042531072256859165199, −3.84756711507265125489847511679, −3.53511996991014255088625359725, −3.48397817706385910086685121560, −3.33346025282642208665222941230, −2.33644919905903927464820475449, −2.24389398766611432919762071086, −2.02406169764490789053824577434, −1.47569405733674198145364353982, −1.37838222334649328233933981507, −0.860939740144385835086401300255, 0.860939740144385835086401300255, 1.37838222334649328233933981507, 1.47569405733674198145364353982, 2.02406169764490789053824577434, 2.24389398766611432919762071086, 2.33644919905903927464820475449, 3.33346025282642208665222941230, 3.48397817706385910086685121560, 3.53511996991014255088625359725, 3.84756711507265125489847511679, 4.54929984042531072256859165199, 4.57896778918635400770656627339, 4.94715596843835069023723653343, 5.11230797549031381038360408317, 5.35898668893587231901003368819, 6.02342953779757556066240162203, 6.24162901151056270447519969965, 6.25115543635079235497105364240, 7.04942233489018741394881458200, 7.12167511553267171037756707280, 7.41716477950073777319290556333, 7.61793210200202970135012910545, 8.037280753225211284468601151860, 8.131878632539373894394789052338, 8.165266536736616001604929017023

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