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Properties

Label 10-4205e5-1.1-c1e5-0-3
Degree $10$
Conductor $1.315\times 10^{18}$
Sign $-1$
Analytic cond. $4.26791\times 10^{7}$
Root an. cond. $5.79457$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 5·5-s + 2·6-s − 3·7-s − 3·8-s − 3·9-s − 5·10-s − 2·11-s + 3·14-s − 10·15-s + 4·16-s − 19·17-s + 3·18-s + 19-s + 6·21-s + 2·22-s + 4·23-s + 6·24-s + 15·25-s + 7·27-s + 10·30-s − 18·31-s + 32-s + 4·33-s + 19·34-s − 15·35-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 2.23·5-s + 0.816·6-s − 1.13·7-s − 1.06·8-s − 9-s − 1.58·10-s − 0.603·11-s + 0.801·14-s − 2.58·15-s + 16-s − 4.60·17-s + 0.707·18-s + 0.229·19-s + 1.30·21-s + 0.426·22-s + 0.834·23-s + 1.22·24-s + 3·25-s + 1.34·27-s + 1.82·30-s − 3.23·31-s + 0.176·32-s + 0.696·33-s + 3.25·34-s − 2.53·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(5^{5} \cdot 29^{10}\)
Sign: $-1$
Analytic conductor: \(4.26791\times 10^{7}\)
Root analytic conductor: \(5.79457\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 5^{5} \cdot 29^{10} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad5$C_1$ \( ( 1 - T )^{5} \)
29 \( 1 \)
good2$C_2 \wr S_5$ \( 1 + T + T^{2} + p^{2} T^{3} + 3 T^{4} + T^{5} + 3 p T^{6} + p^{4} T^{7} + p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 + 2 T + 7 T^{2} + 13 T^{3} + 10 p T^{4} + 55 T^{5} + 10 p^{2} T^{6} + 13 p^{2} T^{7} + 7 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 3 T + 15 T^{2} + 29 T^{3} + 23 p T^{4} + 339 T^{5} + 23 p^{2} T^{6} + 29 p^{2} T^{7} + 15 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 2 T + 38 T^{2} + 42 T^{3} + 659 T^{4} + 507 T^{5} + 659 p T^{6} + 42 p^{2} T^{7} + 38 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 20 T^{2} + 42 T^{3} + 209 T^{4} + 97 p T^{5} + 209 p T^{6} + 42 p^{2} T^{7} + 20 p^{3} T^{8} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 19 T + 220 T^{2} + 1732 T^{3} + 10416 T^{4} + 48237 T^{5} + 10416 p T^{6} + 1732 p^{2} T^{7} + 220 p^{3} T^{8} + 19 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - T + 78 T^{2} - 35 T^{3} + 2627 T^{4} - 625 T^{5} + 2627 p T^{6} - 35 p^{2} T^{7} + 78 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 4 T + 96 T^{2} - 335 T^{3} + 4040 T^{4} - 11267 T^{5} + 4040 p T^{6} - 335 p^{2} T^{7} + 96 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 18 T + 275 T^{2} + 2595 T^{3} + 21254 T^{4} + 126505 T^{5} + 21254 p T^{6} + 2595 p^{2} T^{7} + 275 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 2 T + 80 T^{2} + 375 T^{3} + 2290 T^{4} + 22265 T^{5} + 2290 p T^{6} + 375 p^{2} T^{7} + 80 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 2 T + 109 T^{2} + 269 T^{3} + 6604 T^{4} + 14767 T^{5} + 6604 p T^{6} + 269 p^{2} T^{7} + 109 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 22 T + 350 T^{2} + 3762 T^{3} + 34123 T^{4} + 241123 T^{5} + 34123 p T^{6} + 3762 p^{2} T^{7} + 350 p^{3} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 2 T + 182 T^{2} - 322 T^{3} + 15091 T^{4} - 22153 T^{5} + 15091 p T^{6} - 322 p^{2} T^{7} + 182 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 14 T + 128 T^{2} + 1603 T^{3} + 13482 T^{4} + 85069 T^{5} + 13482 p T^{6} + 1603 p^{2} T^{7} + 128 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 20 T + 384 T^{2} - 4494 T^{3} + 49003 T^{4} - 390251 T^{5} + 49003 p T^{6} - 4494 p^{2} T^{7} + 384 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 5 T + 58 T^{2} + 320 T^{3} + 1846 T^{4} + 40683 T^{5} + 1846 p T^{6} + 320 p^{2} T^{7} + 58 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 21 T + 392 T^{2} + 5058 T^{3} + 54770 T^{4} + 490501 T^{5} + 54770 p T^{6} + 5058 p^{2} T^{7} + 392 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 4 T + 243 T^{2} + 459 T^{3} + 26400 T^{4} + 27539 T^{5} + 26400 p T^{6} + 459 p^{2} T^{7} + 243 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 28 T + 451 T^{2} + 4697 T^{3} + 40258 T^{4} + 321827 T^{5} + 40258 p T^{6} + 4697 p^{2} T^{7} + 451 p^{3} T^{8} + 28 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 3 T + 148 T^{2} + 1016 T^{3} + 14230 T^{4} + 124135 T^{5} + 14230 p T^{6} + 1016 p^{2} T^{7} + 148 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 8 T + 51 T^{2} - 1107 T^{3} + 10766 T^{4} - 49059 T^{5} + 10766 p T^{6} - 1107 p^{2} T^{7} + 51 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 34 T + 481 T^{2} - 1879 T^{3} - 38568 T^{4} + 652859 T^{5} - 38568 p T^{6} - 1879 p^{2} T^{7} + 481 p^{3} T^{8} - 34 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 9 T + 339 T^{2} + 3194 T^{3} + 54021 T^{4} + 452547 T^{5} + 54021 p T^{6} + 3194 p^{2} T^{7} + 339 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.38793340403305927336471181093, −5.36193691472971951280787891314, −5.20008067352100838750521085099, −4.95953019754265346570286813753, −4.90851761830961027198381936671, −4.51299758414817032817752545057, −4.47252696455220729216181350272, −4.40089748220770563737607140320, −4.28500756698698299578505139928, −3.80340182641658453953101075652, −3.51747982214993566318402831087, −3.46462412890037505664121717579, −3.40941524390359957177212550349, −3.12805402784365255079366956767, −3.03673990796860739830719538903, −2.72497896670605168756404664843, −2.53852058217544949438741529867, −2.49792810825375743361365602135, −2.27421235976264044436882644364, −1.91749211285794864864701837129, −1.90017899413792410491872737844, −1.80014133748797939287248327236, −1.40744837257987688700519077899, −1.20282522586832668508859049040, −0.887083036256129725478936855607, 0, 0, 0, 0, 0, 0.887083036256129725478936855607, 1.20282522586832668508859049040, 1.40744837257987688700519077899, 1.80014133748797939287248327236, 1.90017899413792410491872737844, 1.91749211285794864864701837129, 2.27421235976264044436882644364, 2.49792810825375743361365602135, 2.53852058217544949438741529867, 2.72497896670605168756404664843, 3.03673990796860739830719538903, 3.12805402784365255079366956767, 3.40941524390359957177212550349, 3.46462412890037505664121717579, 3.51747982214993566318402831087, 3.80340182641658453953101075652, 4.28500756698698299578505139928, 4.40089748220770563737607140320, 4.47252696455220729216181350272, 4.51299758414817032817752545057, 4.90851761830961027198381936671, 4.95953019754265346570286813753, 5.20008067352100838750521085099, 5.36193691472971951280787891314, 5.38793340403305927336471181093

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