Properties

Degree 10
Conductor $ 2^{5} \cdot 3^{10} \cdot 223^{5} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 5

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·2-s + 15·4-s − 5·5-s − 7-s + 35·8-s − 25·10-s − 9·11-s − 5·14-s + 70·16-s − 6·17-s − 4·19-s − 75·20-s − 45·22-s − 16·23-s + 4·25-s − 15·28-s − 8·29-s − 31-s + 126·32-s − 30·34-s + 5·35-s − 2·37-s − 20·38-s − 175·40-s − 4·41-s + 3·43-s − 135·44-s + ⋯
L(s)  = 1  + 3.53·2-s + 15/2·4-s − 2.23·5-s − 0.377·7-s + 12.3·8-s − 7.90·10-s − 2.71·11-s − 1.33·14-s + 35/2·16-s − 1.45·17-s − 0.917·19-s − 16.7·20-s − 9.59·22-s − 3.33·23-s + 4/5·25-s − 2.83·28-s − 1.48·29-s − 0.179·31-s + 22.2·32-s − 5.14·34-s + 0.845·35-s − 0.328·37-s − 3.24·38-s − 27.6·40-s − 0.624·41-s + 0.457·43-s − 20.3·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{10} \cdot 223^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{10} \cdot 223^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(10\)
\( N \)  =  \(2^{5} \cdot 3^{10} \cdot 223^{5}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4014} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  5
Selberg data  =  $(10,\ 2^{5} \cdot 3^{10} \cdot 223^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;223\}$,\(F_p(T)\) is a polynomial of degree 10. If $p \in \{2,\;3,\;223\}$, then $F_p(T)$ is a polynomial of degree at most 9.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - T )^{5} \)
3 \( 1 \)
223$C_1$ \( ( 1 + T )^{5} \)
good5$C_2 \wr S_5$ \( 1 + p T + 21 T^{2} + 59 T^{3} + 136 T^{4} + 64 p T^{5} + 136 p T^{6} + 59 p^{2} T^{7} + 21 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + T + 17 T^{2} + 13 T^{3} + 184 T^{4} + 164 T^{5} + 184 p T^{6} + 13 p^{2} T^{7} + 17 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 9 T + 61 T^{2} + 314 T^{3} + 1370 T^{4} + 4797 T^{5} + 1370 p T^{6} + 314 p^{2} T^{7} + 61 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 41 T^{2} - 54 T^{3} + 735 T^{4} - 107 p T^{5} + 735 p T^{6} - 54 p^{2} T^{7} + 41 p^{3} T^{8} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 6 T + 80 T^{2} + 341 T^{3} + 2587 T^{4} + 8190 T^{5} + 2587 p T^{6} + 341 p^{2} T^{7} + 80 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 4 T + 79 T^{2} + 216 T^{3} + 2575 T^{4} + 5267 T^{5} + 2575 p T^{6} + 216 p^{2} T^{7} + 79 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 16 T + 182 T^{2} + 1407 T^{3} + 9129 T^{4} + 46862 T^{5} + 9129 p T^{6} + 1407 p^{2} T^{7} + 182 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 8 T + 3 p T^{2} + 554 T^{3} + 3949 T^{4} + 18747 T^{5} + 3949 p T^{6} + 554 p^{2} T^{7} + 3 p^{4} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + T + 101 T^{2} + 143 T^{3} + 5000 T^{4} + 6444 T^{5} + 5000 p T^{6} + 143 p^{2} T^{7} + 101 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 2 T + 72 T^{2} - 51 T^{3} + 3779 T^{4} - 146 T^{5} + 3779 p T^{6} - 51 p^{2} T^{7} + 72 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 4 T + 96 T^{2} - 57 T^{3} + 3551 T^{4} - 12502 T^{5} + 3551 p T^{6} - 57 p^{2} T^{7} + 96 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 3 T + 27 T^{2} + 246 T^{3} + 2882 T^{4} - 10727 T^{5} + 2882 p T^{6} + 246 p^{2} T^{7} + 27 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 18 T + 275 T^{2} + 2774 T^{3} + 24661 T^{4} + 177861 T^{5} + 24661 p T^{6} + 2774 p^{2} T^{7} + 275 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 26 T + 501 T^{2} + 6338 T^{3} + 66181 T^{4} + 524547 T^{5} + 66181 p T^{6} + 6338 p^{2} T^{7} + 501 p^{3} T^{8} + 26 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 21 T + 191 T^{2} + 1174 T^{3} + 7728 T^{4} + 57933 T^{5} + 7728 p T^{6} + 1174 p^{2} T^{7} + 191 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 20 T + 309 T^{2} + 3002 T^{3} + 27263 T^{4} + 201443 T^{5} + 27263 p T^{6} + 3002 p^{2} T^{7} + 309 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 5 T + 207 T^{2} + 1391 T^{3} + 20426 T^{4} + 142580 T^{5} + 20426 p T^{6} + 1391 p^{2} T^{7} + 207 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 17 T + 390 T^{2} + 4187 T^{3} + 54301 T^{4} + 418152 T^{5} + 54301 p T^{6} + 4187 p^{2} T^{7} + 390 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 5 T + 247 T^{2} - 1122 T^{3} + 28636 T^{4} - 113547 T^{5} + 28636 p T^{6} - 1122 p^{2} T^{7} + 247 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 21 T + 385 T^{2} + 3818 T^{3} + 39648 T^{4} + 299687 T^{5} + 39648 p T^{6} + 3818 p^{2} T^{7} + 385 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 11 T + 213 T^{2} + 2511 T^{3} + 31820 T^{4} + 244036 T^{5} + 31820 p T^{6} + 2511 p^{2} T^{7} + 213 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 5 T + 391 T^{2} - 1393 T^{3} + 64134 T^{4} - 168644 T^{5} + 64134 p T^{6} - 1393 p^{2} T^{7} + 391 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 11 T + 347 T^{2} + 3389 T^{3} + 60520 T^{4} + 444664 T^{5} + 60520 p T^{6} + 3389 p^{2} T^{7} + 347 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.20215737038215989748935838975, −5.16203934471662344695036348283, −4.97767703099070636403481815750, −4.96604493781432318476151352138, −4.69249384697693374410123973765, −4.39025823189463432230473701386, −4.34232724473818014368158973394, −4.28668423931572391692316515763, −4.26765570798430688114115023466, −4.18310861404999809440126400088, −3.65366665827196988532709149497, −3.55818512553463386015267206431, −3.54729462559134042702247929972, −3.37955323140548571450300994581, −3.23998008874760753875164178109, −2.97196068572264341363672821454, −2.71371680099437928475709910910, −2.70708682476567549062205826560, −2.45222773144984568089470746235, −2.43329179631741334718209031751, −1.88009574471048767165434544330, −1.78239988901328000640044416830, −1.56700354287063044159040422494, −1.51219697667423181271456436689, −1.49385049171834636435184798567, 0, 0, 0, 0, 0, 1.49385049171834636435184798567, 1.51219697667423181271456436689, 1.56700354287063044159040422494, 1.78239988901328000640044416830, 1.88009574471048767165434544330, 2.43329179631741334718209031751, 2.45222773144984568089470746235, 2.70708682476567549062205826560, 2.71371680099437928475709910910, 2.97196068572264341363672821454, 3.23998008874760753875164178109, 3.37955323140548571450300994581, 3.54729462559134042702247929972, 3.55818512553463386015267206431, 3.65366665827196988532709149497, 4.18310861404999809440126400088, 4.26765570798430688114115023466, 4.28668423931572391692316515763, 4.34232724473818014368158973394, 4.39025823189463432230473701386, 4.69249384697693374410123973765, 4.96604493781432318476151352138, 4.97767703099070636403481815750, 5.16203934471662344695036348283, 5.20215737038215989748935838975

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