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Properties

Label 10-1143e5-1.1-c1e5-0-0
Degree $10$
Conductor $1.951\times 10^{15}$
Sign $1$
Analytic cond. $63331.0$
Root an. cond. $3.02107$
Motivic weight $1$
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  + 2-s − 4·4-s + 5·5-s − 4·8-s + 5·10-s + 16·11-s + 3·13-s + 6·16-s + 6·17-s − 8·19-s − 20·20-s + 16·22-s + 9·23-s + 2·25-s + 3·26-s + 17·29-s − 9·31-s + 5·32-s + 6·34-s − 37-s − 8·38-s − 20·40-s + 2·41-s − 4·43-s − 64·44-s + 9·46-s + 8·47-s + ⋯
L(s)  = 1  + 0.707·2-s − 2·4-s + 2.23·5-s − 1.41·8-s + 1.58·10-s + 4.82·11-s + 0.832·13-s + 3/2·16-s + 1.45·17-s − 1.83·19-s − 4.47·20-s + 3.41·22-s + 1.87·23-s + 2/5·25-s + 0.588·26-s + 3.15·29-s − 1.61·31-s + 0.883·32-s + 1.02·34-s − 0.164·37-s − 1.29·38-s − 3.16·40-s + 0.312·41-s − 0.609·43-s − 9.64·44-s + 1.32·46-s + 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 127^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 127^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{10} \cdot 127^{5}\)
Sign: $1$
Analytic conductor: \(63331.0\)
Root analytic conductor: \(3.02107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 3^{10} \cdot 127^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(12.23561392\)
\(L(\frac12)\) \(\approx\) \(12.23561392\)
\(L(\frac{3}{2})\) 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 \( 1 \)
127$C_1$ \( ( 1 + T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - T + 5 T^{2} - 5 T^{3} + 15 T^{4} - 7 p T^{5} + 15 p T^{6} - 5 p^{2} T^{7} + 5 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 - p T + 23 T^{2} - 76 T^{3} + 44 p T^{4} - 526 T^{5} + 44 p^{2} T^{6} - 76 p^{2} T^{7} + 23 p^{3} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 11 T^{2} + 8 T^{3} + 66 T^{4} + 48 T^{5} + 66 p T^{6} + 8 p^{2} T^{7} + 11 p^{3} T^{8} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 16 T + 146 T^{2} - 84 p T^{3} + 4406 T^{4} - 16458 T^{5} + 4406 p T^{6} - 84 p^{3} T^{7} + 146 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 3 T + 54 T^{2} - 109 T^{3} + 1220 T^{4} - 1821 T^{5} + 1220 p T^{6} - 109 p^{2} T^{7} + 54 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 6 T + 78 T^{2} - 346 T^{3} + 2542 T^{4} - 8458 T^{5} + 2542 p T^{6} - 346 p^{2} T^{7} + 78 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 8 T + 68 T^{2} + 360 T^{3} + 2244 T^{4} + 9408 T^{5} + 2244 p T^{6} + 360 p^{2} T^{7} + 68 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 9 T + 5 p T^{2} - 788 T^{3} + 5266 T^{4} - 26742 T^{5} + 5266 p T^{6} - 788 p^{2} T^{7} + 5 p^{4} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 17 T + 205 T^{2} - 1736 T^{3} + 12750 T^{4} - 74030 T^{5} + 12750 p T^{6} - 1736 p^{2} T^{7} + 205 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 9 T + 100 T^{2} + 597 T^{3} + 4624 T^{4} + 23977 T^{5} + 4624 p T^{6} + 597 p^{2} T^{7} + 100 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + T + 118 T^{2} + 339 T^{3} + 6252 T^{4} + 22331 T^{5} + 6252 p T^{6} + 339 p^{2} T^{7} + 118 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 2 T + 54 T^{2} + 202 T^{3} + 1422 T^{4} + 16782 T^{5} + 1422 p T^{6} + 202 p^{2} T^{7} + 54 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 4 T + 151 T^{2} + 808 T^{3} + 10346 T^{4} + 54440 T^{5} + 10346 p T^{6} + 808 p^{2} T^{7} + 151 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 8 T + 114 T^{2} - 444 T^{3} + 4134 T^{4} - 7366 T^{5} + 4134 p T^{6} - 444 p^{2} T^{7} + 114 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 15 T + 233 T^{2} - 1932 T^{3} + 17170 T^{4} - 112314 T^{5} + 17170 p T^{6} - 1932 p^{2} T^{7} + 233 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 19 T + 357 T^{2} - 4092 T^{3} + 44664 T^{4} - 352802 T^{5} + 44664 p T^{6} - 4092 p^{2} T^{7} + 357 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - T + 242 T^{2} - 315 T^{3} + 26064 T^{4} - 30671 T^{5} + 26064 p T^{6} - 315 p^{2} T^{7} + 242 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 2 T + 167 T^{2} + 864 T^{3} + 13842 T^{4} + 94396 T^{5} + 13842 p T^{6} + 864 p^{2} T^{7} + 167 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 2 p T^{2} - 652 T^{3} + 15022 T^{4} - 47082 T^{5} + 15022 p T^{6} - 652 p^{2} T^{7} + 2 p^{4} T^{8} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 13 T + 130 T^{2} - 2387 T^{3} + 23112 T^{4} - 158859 T^{5} + 23112 p T^{6} - 2387 p^{2} T^{7} + 130 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 28 T + 580 T^{2} + 8324 T^{3} + 101500 T^{4} + 975704 T^{5} + 101500 p T^{6} + 8324 p^{2} T^{7} + 580 p^{3} T^{8} + 28 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - T + 175 T^{2} - 196 T^{3} + 13146 T^{4} - 25174 T^{5} + 13146 p T^{6} - 196 p^{2} T^{7} + 175 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + T + 177 T^{2} + 344 T^{3} + 16950 T^{4} + 56734 T^{5} + 16950 p T^{6} + 344 p^{2} T^{7} + 177 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 28 T + 405 T^{2} - 4800 T^{3} + 63802 T^{4} - 725992 T^{5} + 63802 p T^{6} - 4800 p^{2} T^{7} + 405 p^{3} T^{8} - 28 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.81236791651547885134306694424, −5.77790215072254009820905041215, −5.49966154695864397403655859100, −5.44254867458881017978859596805, −5.29714724964244368544308667566, −4.79496503759200843834659750692, −4.71787636780856480687064181789, −4.68865500185768122398976466647, −4.44537529206607728372071914071, −4.26449022435188196528182497158, −3.91189389318844833214025754662, −3.80309501443659086133900352262, −3.79517104157041291262710177994, −3.58771695569346908614477976712, −3.41031719193765221419338716851, −3.14011642276793315234147660860, −2.59368111037658492692217040600, −2.39952147970314901155315614641, −2.33837241771796924210036028523, −1.74961670760768143442559097165, −1.65840649958548290235169214077, −1.40851084851587041099492579796, −1.11529937598930375848123377642, −0.934638777988258646526324652742, −0.57366396375720515251191096333, 0.57366396375720515251191096333, 0.934638777988258646526324652742, 1.11529937598930375848123377642, 1.40851084851587041099492579796, 1.65840649958548290235169214077, 1.74961670760768143442559097165, 2.33837241771796924210036028523, 2.39952147970314901155315614641, 2.59368111037658492692217040600, 3.14011642276793315234147660860, 3.41031719193765221419338716851, 3.58771695569346908614477976712, 3.79517104157041291262710177994, 3.80309501443659086133900352262, 3.91189389318844833214025754662, 4.26449022435188196528182497158, 4.44537529206607728372071914071, 4.68865500185768122398976466647, 4.71787636780856480687064181789, 4.79496503759200843834659750692, 5.29714724964244368544308667566, 5.44254867458881017978859596805, 5.49966154695864397403655859100, 5.77790215072254009820905041215, 5.81236791651547885134306694424

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