Properties

Degree 10
Conductor $ 2^{5} \cdot 3^{5} \cdot 17^{5} \cdot 59^{5} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s + 5·3-s + 15·4-s − 5-s − 25·6-s − 7-s − 35·8-s + 15·9-s + 5·10-s + 6·11-s + 75·12-s − 2·13-s + 5·14-s − 5·15-s + 70·16-s − 5·17-s − 75·18-s + 4·19-s − 15·20-s − 5·21-s − 30·22-s + 12·23-s − 175·24-s − 10·25-s + 10·26-s + 35·27-s − 15·28-s + ⋯
L(s)  = 1  − 3.53·2-s + 2.88·3-s + 15/2·4-s − 0.447·5-s − 10.2·6-s − 0.377·7-s − 12.3·8-s + 5·9-s + 1.58·10-s + 1.80·11-s + 21.6·12-s − 0.554·13-s + 1.33·14-s − 1.29·15-s + 35/2·16-s − 1.21·17-s − 17.6·18-s + 0.917·19-s − 3.35·20-s − 1.09·21-s − 6.39·22-s + 2.50·23-s − 35.7·24-s − 2·25-s + 1.96·26-s + 6.73·27-s − 2.83·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 17^{5} \cdot 59^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 17^{5} \cdot 59^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(10\)
\( N \)  =  \(2^{5} \cdot 3^{5} \cdot 17^{5} \cdot 59^{5}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6018} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(10,\ 2^{5} \cdot 3^{5} \cdot 17^{5} \cdot 59^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $13.72274697$
$L(\frac12)$  $\approx$  $13.72274697$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;17,\;59\}$, \(F_p\) is a polynomial of degree 10. If $p \in \{2,\;3,\;17,\;59\}$, then $F_p$ is a polynomial of degree at most 9.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( ( 1 + T )^{5} \)
3$C_1$ \( ( 1 - T )^{5} \)
17$C_1$ \( ( 1 + T )^{5} \)
59$C_1$ \( ( 1 + T )^{5} \)
good5$C_2 \wr S_5$ \( 1 + T + 11 T^{2} + 2 p T^{3} + 3 p^{2} T^{4} + 82 T^{5} + 3 p^{3} T^{6} + 2 p^{3} T^{7} + 11 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + T + 24 T^{2} + 27 T^{3} + 276 T^{4} + 282 T^{5} + 276 p T^{6} + 27 p^{2} T^{7} + 24 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 6 T + 37 T^{2} - 177 T^{3} + 739 T^{4} - 2490 T^{5} + 739 p T^{6} - 177 p^{2} T^{7} + 37 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 2 T + 42 T^{2} + 66 T^{3} + 918 T^{4} + 1182 T^{5} + 918 p T^{6} + 66 p^{2} T^{7} + 42 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 4 T + 81 T^{2} - 240 T^{3} + 147 p T^{4} - 6276 T^{5} + 147 p^{2} T^{6} - 240 p^{2} T^{7} + 81 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 12 T + 127 T^{2} - 867 T^{3} + 5287 T^{4} - 26424 T^{5} + 5287 p T^{6} - 867 p^{2} T^{7} + 127 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 19 T + 215 T^{2} - 1792 T^{3} + 12537 T^{4} - 73546 T^{5} + 12537 p T^{6} - 1792 p^{2} T^{7} + 215 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 5 T + 99 T^{2} - 300 T^{3} + 4311 T^{4} - 9180 T^{5} + 4311 p T^{6} - 300 p^{2} T^{7} + 99 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 11 T + 165 T^{2} + 1122 T^{3} + 10035 T^{4} + 51894 T^{5} + 10035 p T^{6} + 1122 p^{2} T^{7} + 165 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 6 T + 187 T^{2} - 849 T^{3} + 353 p T^{4} - 49488 T^{5} + 353 p^{2} T^{6} - 849 p^{2} T^{7} + 187 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 2 T + 93 T^{2} - 81 T^{3} + 4347 T^{4} - 1014 T^{5} + 4347 p T^{6} - 81 p^{2} T^{7} + 93 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 22 T + 293 T^{2} - 3031 T^{3} + 25605 T^{4} - 183682 T^{5} + 25605 p T^{6} - 3031 p^{2} T^{7} + 293 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 15 T + 292 T^{2} - 2799 T^{3} + 31312 T^{4} - 213858 T^{5} + 31312 p T^{6} - 2799 p^{2} T^{7} + 292 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 16 T + 321 T^{2} - 3165 T^{3} + 36369 T^{4} - 261786 T^{5} + 36369 p T^{6} - 3165 p^{2} T^{7} + 321 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 25 T + 312 T^{2} - 1953 T^{3} + 5250 T^{4} + 8760 T^{5} + 5250 p T^{6} - 1953 p^{2} T^{7} + 312 p^{3} T^{8} - 25 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 259 T^{2} + 243 T^{3} + 30943 T^{4} + 31680 T^{5} + 30943 p T^{6} + 243 p^{2} T^{7} + 259 p^{3} T^{8} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 10 T + 189 T^{2} + 1767 T^{3} + 17205 T^{4} + 159624 T^{5} + 17205 p T^{6} + 1767 p^{2} T^{7} + 189 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 10 T + 201 T^{2} - 1224 T^{3} + 16233 T^{4} - 68586 T^{5} + 16233 p T^{6} - 1224 p^{2} T^{7} + 201 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 19 T + 455 T^{2} - 5380 T^{3} + 74511 T^{4} - 630862 T^{5} + 74511 p T^{6} - 5380 p^{2} T^{7} + 455 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 23 T + 563 T^{2} - 8012 T^{3} + 108639 T^{4} - 1058708 T^{5} + 108639 p T^{6} - 8012 p^{2} T^{7} + 563 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 8 T + 468 T^{2} - 2808 T^{3} + 88296 T^{4} - 393462 T^{5} + 88296 p T^{6} - 2808 p^{2} T^{7} + 468 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.73272109022745040524684943752, −4.39390805266223494472711229883, −4.31772639145234270503273096179, −4.15312137529228180420744009098, −4.07053221144089297426757594710, −3.66803928775259819632959632962, −3.65005391764941215143034594835, −3.53509343352594899718176506426, −3.35200576990292072830294597469, −3.21893926815618757636543110178, −2.94736205863431346984677098465, −2.79286037934890520554847767591, −2.59477830679039228778397139738, −2.57684737901484364137224665864, −2.52315163885319717623102513985, −1.99400511970926957029571025385, −1.88422329340514350656408090316, −1.87683752195638375851255565029, −1.78325398793599806838254188805, −1.55336074914897799457765500156, −0.946402405965776991617956002487, −0.801537594444080251525370606947, −0.77731502209312401205209944972, −0.71196915385092737824283181565, −0.57390895122571145577143980477, 0.57390895122571145577143980477, 0.71196915385092737824283181565, 0.77731502209312401205209944972, 0.801537594444080251525370606947, 0.946402405965776991617956002487, 1.55336074914897799457765500156, 1.78325398793599806838254188805, 1.87683752195638375851255565029, 1.88422329340514350656408090316, 1.99400511970926957029571025385, 2.52315163885319717623102513985, 2.57684737901484364137224665864, 2.59477830679039228778397139738, 2.79286037934890520554847767591, 2.94736205863431346984677098465, 3.21893926815618757636543110178, 3.35200576990292072830294597469, 3.53509343352594899718176506426, 3.65005391764941215143034594835, 3.66803928775259819632959632962, 4.07053221144089297426757594710, 4.15312137529228180420744009098, 4.31772639145234270503273096179, 4.39390805266223494472711229883, 4.73272109022745040524684943752

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