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Properties

Label 10-2576e5-1.1-c1e5-0-3
Degree $10$
Conductor $1.134\times 10^{17}$
Sign $-1$
Analytic cond. $3.68226\times 10^{6}$
Root an. cond. $4.53535$
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  − 4·5-s − 5·7-s − 2·9-s + 4·11-s − 6·13-s − 12·17-s − 6·19-s + 5·23-s + 5·25-s − 4·29-s − 30·31-s + 20·35-s + 4·37-s + 6·41-s + 12·43-s + 8·45-s − 10·47-s + 15·49-s + 16·53-s − 16·55-s − 22·59-s − 18·61-s + 10·63-s + 24·65-s + 2·67-s − 4·71-s − 2·73-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.88·7-s − 2/3·9-s + 1.20·11-s − 1.66·13-s − 2.91·17-s − 1.37·19-s + 1.04·23-s + 25-s − 0.742·29-s − 5.38·31-s + 3.38·35-s + 0.657·37-s + 0.937·41-s + 1.82·43-s + 1.19·45-s − 1.45·47-s + 15/7·49-s + 2.19·53-s − 2.15·55-s − 2.86·59-s − 2.30·61-s + 1.25·63-s + 2.97·65-s + 0.244·67-s − 0.474·71-s − 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{20} \cdot 7^{5} \cdot 23^{5}\)
Sign: $-1$
Analytic conductor: \(3.68226\times 10^{6}\)
Root analytic conductor: \(4.53535\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{20} \cdot 7^{5} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad2 \( 1 \)
7$C_1$ \( ( 1 + T )^{5} \)
23$C_1$ \( ( 1 - T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + 2 T^{2} + 11 T^{4} - 10 T^{5} + 11 p T^{6} + 2 p^{3} T^{8} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 4 T + 11 T^{2} + 26 T^{3} + 92 T^{4} + 228 T^{5} + 92 p T^{6} + 26 p^{2} T^{7} + 11 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.68211541834735042003155932556, −5.50906612457544477269276395043, −5.44752647026521909305605765621, −5.35217938106872138756259176811, −4.99924908897728651042410998743, −4.67677281147919535997017088868, −4.64969656781408981710177619461, −4.39089215280688668434137673731, −4.29439035359005666118015336051, −4.09493320769147214868816913164, −4.09350911448998470927197000474, −3.90721326189091888763918127416, −3.65457266260968109349113223883, −3.47063248802729021150324702716, −3.29692880438543948317520096813, −3.13966715963712556143321168372, −2.87000816727461361356298438374, −2.67568458025126800538962654152, −2.37240294059899766754901060629, −2.37226592151009861572973674460, −2.23057718972739939331513908560, −1.84062590292578951321539690057, −1.45843665984954440644777906161, −1.35048514239894117132309712368, −1.08319853328756332988166989997, 0, 0, 0, 0, 0, 1.08319853328756332988166989997, 1.35048514239894117132309712368, 1.45843665984954440644777906161, 1.84062590292578951321539690057, 2.23057718972739939331513908560, 2.37226592151009861572973674460, 2.37240294059899766754901060629, 2.67568458025126800538962654152, 2.87000816727461361356298438374, 3.13966715963712556143321168372, 3.29692880438543948317520096813, 3.47063248802729021150324702716, 3.65457266260968109349113223883, 3.90721326189091888763918127416, 4.09350911448998470927197000474, 4.09493320769147214868816913164, 4.29439035359005666118015336051, 4.39089215280688668434137673731, 4.64969656781408981710177619461, 4.67677281147919535997017088868, 4.99924908897728651042410998743, 5.35217938106872138756259176811, 5.44752647026521909305605765621, 5.50906612457544477269276395043, 5.68211541834735042003155932556

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