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Properties

Label 10-3381e5-1.1-c1e5-0-1
Degree $10$
Conductor $4.418\times 10^{17}$
Sign $-1$
Analytic cond. $1.43420\times 10^{7}$
Root an. cond. $5.19590$
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 − 5·3-s − 2·5-s − 5·6-s + 15·9-s − 2·10-s − 2·11-s − 4·13-s + 10·15-s − 4·16-s − 2·17-s + 15·18-s − 8·19-s − 2·22-s + 5·23-s − 8·25-s − 4·26-s − 35·27-s + 4·29-s + 10·30-s − 12·31-s − 4·32-s + 10·33-s − 2·34-s − 6·37-s − 8·38-s + 20·39-s + ⋯
L(s)  = 1  + 0.707·2-s − 2.88·3-s − 0.894·5-s − 2.04·6-s + 5·9-s − 0.632·10-s − 0.603·11-s − 1.10·13-s + 2.58·15-s − 16-s − 0.485·17-s + 3.53·18-s − 1.83·19-s − 0.426·22-s + 1.04·23-s − 8/5·25-s − 0.784·26-s − 6.73·27-s + 0.742·29-s + 1.82·30-s − 2.15·31-s − 0.707·32-s + 1.74·33-s − 0.342·34-s − 0.986·37-s − 1.29·38-s + 3.20·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 7^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 7^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{5} \cdot 7^{10} \cdot 23^{5}\)
Sign: $-1$
Analytic conductor: \(1.43420\times 10^{7}\)
Root analytic conductor: \(5.19590\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 3^{5} \cdot 7^{10} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad3$C_1$ \( ( 1 + T )^{5} \)
7 \( 1 \)
23$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - T + T^{2} - T^{3} + 5 T^{4} - 5 T^{5} + 5 p T^{6} - p^{2} T^{7} + p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 2 T + 12 T^{2} + 32 T^{3} + 108 T^{4} + 184 T^{5} + 108 p T^{6} + 32 p^{2} T^{7} + 12 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 2 T + 39 T^{2} + 62 T^{3} + 689 T^{4} + 884 T^{5} + 689 p T^{6} + 62 p^{2} T^{7} + 39 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 4 T + 48 T^{2} + 8 p T^{3} + 882 T^{4} + 1312 T^{5} + 882 p T^{6} + 8 p^{3} T^{7} + 48 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 2 T + 47 T^{2} + 214 T^{3} + 925 T^{4} + 6116 T^{5} + 925 p T^{6} + 214 p^{2} T^{7} + 47 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 8 T + 81 T^{2} + 416 T^{3} + 2657 T^{4} + 10368 T^{5} + 2657 p T^{6} + 416 p^{2} T^{7} + 81 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 4 T + 3 p T^{2} - 262 T^{3} + 3845 T^{4} - 9630 T^{5} + 3845 p T^{6} - 262 p^{2} T^{7} + 3 p^{4} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 12 T + 195 T^{2} + 48 p T^{3} + 13173 T^{4} + 69044 T^{5} + 13173 p T^{6} + 48 p^{3} T^{7} + 195 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 6 T + 41 T^{2} + 240 T^{3} + 25 p T^{4} + 998 T^{5} + 25 p^{2} T^{6} + 240 p^{2} T^{7} + 41 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 6 T + 153 T^{2} + 558 T^{3} + 9537 T^{4} + 25136 T^{5} + 9537 p T^{6} + 558 p^{2} T^{7} + 153 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 24 T + 422 T^{2} + 4844 T^{3} + 45704 T^{4} + 326508 T^{5} + 45704 p T^{6} + 4844 p^{2} T^{7} + 422 p^{3} T^{8} + 24 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 24 T + 371 T^{2} + 4160 T^{3} + 37490 T^{4} + 279632 T^{5} + 37490 p T^{6} + 4160 p^{2} T^{7} + 371 p^{3} T^{8} + 24 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 2 T + 142 T^{2} + 136 T^{3} + 10004 T^{4} + 18098 T^{5} + 10004 p T^{6} + 136 p^{2} T^{7} + 142 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 16 T + 324 T^{2} + 3466 T^{3} + 39818 T^{4} + 298096 T^{5} + 39818 p T^{6} + 3466 p^{2} T^{7} + 324 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 22 T + 324 T^{2} + 3642 T^{3} + 37164 T^{4} + 311164 T^{5} + 37164 p T^{6} + 3642 p^{2} T^{7} + 324 p^{3} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 16 T + 252 T^{2} + 2624 T^{3} + 25242 T^{4} + 206884 T^{5} + 25242 p T^{6} + 2624 p^{2} T^{7} + 252 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 16 T + 410 T^{2} - 4390 T^{3} + 61590 T^{4} - 463028 T^{5} + 61590 p T^{6} - 4390 p^{2} T^{7} + 410 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 37 T^{2} - 576 T^{3} + 6573 T^{4} + 16504 T^{5} + 6573 p T^{6} - 576 p^{2} T^{7} + 37 p^{3} T^{8} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 12 T + 7 T^{2} + 1020 T^{3} + 4797 T^{4} - 124872 T^{5} + 4797 p T^{6} + 1020 p^{2} T^{7} + 7 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 10 T + 397 T^{2} + 3146 T^{3} + 64833 T^{4} + 384240 T^{5} + 64833 p T^{6} + 3146 p^{2} T^{7} + 397 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 16 T + 236 T^{2} - 3674 T^{3} + 41164 T^{4} - 393112 T^{5} + 41164 p T^{6} - 3674 p^{2} T^{7} + 236 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 4 T + 383 T^{2} - 1896 T^{3} + 64133 T^{4} - 292516 T^{5} + 64133 p T^{6} - 1896 p^{2} T^{7} + 383 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.58175612232316179751977982337, −5.16825306238254259905780614099, −5.14097316226701265655770015584, −4.92351191810153062431118745760, −4.91648539010852432027830761268, −4.77263780449089704507687898525, −4.67427494068115249679561885040, −4.47177202061132508921263855758, −4.40020310846032748303835121093, −4.30735087486345841716469007482, −3.74886845654443143788209730877, −3.67117293685936556768412915123, −3.61652186230194869796192569441, −3.48725410505116672154394870218, −3.35276844791403081498030420103, −3.06197372247463101654167557982, −2.66561148352100372822327825150, −2.59369197473034699007243650489, −2.28229845248746739465431293714, −1.95133603143119443259183780877, −1.79665209201439800413501987725, −1.78958832044571937198890335764, −1.41127207776529534118231471253, −1.35059434317125230319504956231, −0.868391250688133976037426362506, 0, 0, 0, 0, 0, 0.868391250688133976037426362506, 1.35059434317125230319504956231, 1.41127207776529534118231471253, 1.78958832044571937198890335764, 1.79665209201439800413501987725, 1.95133603143119443259183780877, 2.28229845248746739465431293714, 2.59369197473034699007243650489, 2.66561148352100372822327825150, 3.06197372247463101654167557982, 3.35276844791403081498030420103, 3.48725410505116672154394870218, 3.61652186230194869796192569441, 3.67117293685936556768412915123, 3.74886845654443143788209730877, 4.30735087486345841716469007482, 4.40020310846032748303835121093, 4.47177202061132508921263855758, 4.67427494068115249679561885040, 4.77263780449089704507687898525, 4.91648539010852432027830761268, 4.92351191810153062431118745760, 5.14097316226701265655770015584, 5.16825306238254259905780614099, 5.58175612232316179751977982337

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