Properties

Degree 10
Conductor $ 5^{10} \cdot 7^{5} \cdot 23^{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  − 2·2-s + 3·4-s − 5·7-s − 3·8-s − 2·9-s − 4·11-s + 6·13-s + 10·14-s − 16-s + 12·17-s + 4·18-s + 6·19-s + 8·22-s + 5·23-s − 12·26-s − 15·28-s − 4·29-s + 30·31-s + 5·32-s − 24·34-s − 6·36-s − 4·37-s − 12·38-s + 6·41-s + 12·43-s − 12·44-s − 10·46-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.88·7-s − 1.06·8-s − 2/3·9-s − 1.20·11-s + 1.66·13-s + 2.67·14-s − 1/4·16-s + 2.91·17-s + 0.942·18-s + 1.37·19-s + 1.70·22-s + 1.04·23-s − 2.35·26-s − 2.83·28-s − 0.742·29-s + 5.38·31-s + 0.883·32-s − 4.11·34-s − 36-s − 0.657·37-s − 1.94·38-s + 0.937·41-s + 1.82·43-s − 1.80·44-s − 1.47·46-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(10\)
\( N \)  =  \(5^{10} \cdot 7^{5} \cdot 23^{5}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4025} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(10,\ 5^{10} \cdot 7^{5} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $2.446388343$
$L(\frac12)$  $\approx$  $2.446388343$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{5,\;7,\;23\}$,\(F_p(T)\) is a polynomial of degree 10. If $p \in \{5,\;7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 9.
$p$$\Gal(F_p)$$F_p(T)$
bad5 \( 1 \)
7$C_1$ \( ( 1 + T )^{5} \)
23$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 + p T + T^{2} - T^{3} + p T^{4} + 7 T^{5} + p^{2} T^{6} - p^{2} T^{7} + p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 + 2 T^{2} + 11 T^{4} - 10 T^{5} + 11 p T^{6} + 2 p^{3} T^{8} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 4 T + 27 T^{2} + 28 T^{3} + 126 T^{4} - 400 T^{5} + 126 p T^{6} + 28 p^{2} T^{7} + 27 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 6 T + 56 T^{2} - 266 T^{3} + 1351 T^{4} - 4944 T^{5} + 1351 p T^{6} - 266 p^{2} T^{7} + 56 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 12 T + 91 T^{2} - 430 T^{3} + 1692 T^{4} - 6148 T^{5} + 1692 p T^{6} - 430 p^{2} T^{7} + 91 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 6 T + 67 T^{2} - 360 T^{3} + 2334 T^{4} - 9220 T^{5} + 2334 p T^{6} - 360 p^{2} T^{7} + 67 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 4 T + 34 T^{2} + 214 T^{3} + 1217 T^{4} + 4232 T^{5} + 1217 p T^{6} + 214 p^{2} T^{7} + 34 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 30 T + 502 T^{2} - 5646 T^{3} + 46995 T^{4} - 297598 T^{5} + 46995 p T^{6} - 5646 p^{2} T^{7} + 502 p^{3} T^{8} - 30 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 4 T + 109 T^{2} + 216 T^{3} + 5126 T^{4} + 5064 T^{5} + 5126 p T^{6} + 216 p^{2} T^{7} + 109 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 6 T + 176 T^{2} - 838 T^{3} + 13295 T^{4} - 49000 T^{5} + 13295 p T^{6} - 838 p^{2} T^{7} + 176 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 12 T + 191 T^{2} - 1696 T^{3} + 16226 T^{4} - 101736 T^{5} + 16226 p T^{6} - 1696 p^{2} T^{7} + 191 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 10 T + 110 T^{2} - 38 T^{3} - 3609 T^{4} - 58894 T^{5} - 3609 p T^{6} - 38 p^{2} T^{7} + 110 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 16 T + 317 T^{2} + 3240 T^{3} + 35942 T^{4} + 254032 T^{5} + 35942 p T^{6} + 3240 p^{2} T^{7} + 317 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 22 T + 7 p T^{2} - 5194 T^{3} + 54600 T^{4} - 458288 T^{5} + 54600 p T^{6} - 5194 p^{2} T^{7} + 7 p^{4} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 18 T + 339 T^{2} + 3954 T^{3} + 43744 T^{4} + 348488 T^{5} + 43744 p T^{6} + 3954 p^{2} T^{7} + 339 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 2 T + 35 T^{2} + 204 T^{3} + 6790 T^{4} - 16644 T^{5} + 6790 p T^{6} + 204 p^{2} T^{7} + 35 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 4 T + 254 T^{2} - 860 T^{3} + 30833 T^{4} - 86976 T^{5} + 30833 p T^{6} - 860 p^{2} T^{7} + 254 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 2 T + 168 T^{2} - 946 T^{3} + 18175 T^{4} - 89144 T^{5} + 18175 p T^{6} - 946 p^{2} T^{7} + 168 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 30 T + 703 T^{2} - 10676 T^{3} + 136374 T^{4} - 1310412 T^{5} + 136374 p T^{6} - 10676 p^{2} T^{7} + 703 p^{3} T^{8} - 30 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 8 T + 359 T^{2} + 2224 T^{3} + 55778 T^{4} + 264336 T^{5} + 55778 p T^{6} + 2224 p^{2} T^{7} + 359 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 20 T + 511 T^{2} + 6422 T^{3} + 92672 T^{4} + 821572 T^{5} + 92672 p T^{6} + 6422 p^{2} T^{7} + 511 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 12 T + 371 T^{2} - 3294 T^{3} + 61432 T^{4} - 417340 T^{5} + 61432 p T^{6} - 3294 p^{2} T^{7} + 371 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.97833642844125336051173883878, −4.94229437287290300332728074979, −4.82327374701349192620961318618, −4.49966034321092593205799779576, −4.31339139682242642499576689476, −4.09004884747658101621867722359, −3.99592888680963863602200642469, −3.62424003567001716283429204942, −3.61292867385496940735518057142, −3.32154930869442125169186800964, −3.23888884765524645645014267858, −3.11881927544921843405647070031, −2.85076642811841786055464216772, −2.81679252097921039293356143089, −2.56790214007102384837815112914, −2.53370932152700804856023121755, −2.33091576500480927815931154436, −1.92843561491559684228218792398, −1.59302858438911053648773445340, −1.33821148499433837455629140504, −1.18483011812085461729716526493, −0.976402381734085407172900394593, −0.861909461123922764920358436227, −0.53403699993518861149951819962, −0.28680288490728902721378591750, 0.28680288490728902721378591750, 0.53403699993518861149951819962, 0.861909461123922764920358436227, 0.976402381734085407172900394593, 1.18483011812085461729716526493, 1.33821148499433837455629140504, 1.59302858438911053648773445340, 1.92843561491559684228218792398, 2.33091576500480927815931154436, 2.53370932152700804856023121755, 2.56790214007102384837815112914, 2.81679252097921039293356143089, 2.85076642811841786055464216772, 3.11881927544921843405647070031, 3.23888884765524645645014267858, 3.32154930869442125169186800964, 3.61292867385496940735518057142, 3.62424003567001716283429204942, 3.99592888680963863602200642469, 4.09004884747658101621867722359, 4.31339139682242642499576689476, 4.49966034321092593205799779576, 4.82327374701349192620961318618, 4.94229437287290300332728074979, 4.97833642844125336051173883878

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