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Properties

Label 10-3703e5-1.1-c1e5-0-3
Degree $10$
Conductor $6.963\times 10^{17}$
Sign $1$
Analytic cond. $2.26023\times 10^{7}$
Root an. cond. $5.43770$
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  + 2·2-s + 3·4-s + 4·5-s − 5·7-s + 3·8-s − 2·9-s + 8·10-s + 4·11-s − 6·13-s − 10·14-s − 16-s + 12·17-s − 4·18-s − 6·19-s + 12·20-s + 8·22-s + 5·25-s − 12·26-s − 15·28-s − 4·29-s + 30·31-s − 5·32-s + 24·34-s − 20·35-s − 6·36-s − 4·37-s − 12·38-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.78·5-s − 1.88·7-s + 1.06·8-s − 2/3·9-s + 2.52·10-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 + 2.68·20-s + 1.70·22-s + 25-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 − 3.38·35-s − 36-s − 0.657·37-s − 1.94·38-s + ⋯

Functional equation

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

Invariants

Degree: \(10\)
Conductor: \(7^{5} \cdot 23^{10}\)
Sign: $1$
Analytic conductor: \(2.26023\times 10^{7}\)
Root analytic conductor: \(5.43770\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 7^{5} \cdot 23^{10} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(27.71719519\)
\(L(\frac12)\) \(\approx\) \(27.71719519\)
\(L(\frac{3}{2})\) 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)$
bad7$C_1$ \( ( 1 + T )^{5} \)
23 \( 1 \)
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$C_2 \wr S_5$ \( 1 + 2 T^{2} + 11 T^{4} + 10 T^{5} + 11 p T^{6} + 2 p^{3} T^{8} + p^{5} T^{10} \)
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

−4.98388203679910638046775629902, −4.85739755652325919881449881398, −4.83200494445930651942375990170, −4.59954616608771910130714916585, −4.27924894718961789803270755918, −4.18615125900385369287888021913, −4.13684167405699751017445883358, −3.94681543698952772208780355495, −3.75818639641609945364016092285, −3.51825246183002161730371499957, −3.31589483459575470977295565774, −3.08304345609846962134591971943, −2.98817161039328174894878166289, −2.68011728261128694180907430635, −2.65076253589591555869552374864, −2.53063943221866176894340547775, −2.36796953083227866392770863048, −2.32499858877576992316063063980, −1.93978242527522802039228711509, −1.49952326248333629626661814388, −1.46269843156435704238460480160, −1.23380101175844379715256838489, −0.69732722145957386587436451255, −0.64953836160440086840436403296, −0.45787216697059175790896577780, 0.45787216697059175790896577780, 0.64953836160440086840436403296, 0.69732722145957386587436451255, 1.23380101175844379715256838489, 1.46269843156435704238460480160, 1.49952326248333629626661814388, 1.93978242527522802039228711509, 2.32499858877576992316063063980, 2.36796953083227866392770863048, 2.53063943221866176894340547775, 2.65076253589591555869552374864, 2.68011728261128694180907430635, 2.98817161039328174894878166289, 3.08304345609846962134591971943, 3.31589483459575470977295565774, 3.51825246183002161730371499957, 3.75818639641609945364016092285, 3.94681543698952772208780355495, 4.13684167405699751017445883358, 4.18615125900385369287888021913, 4.27924894718961789803270755918, 4.59954616608771910130714916585, 4.83200494445930651942375990170, 4.85739755652325919881449881398, 4.98388203679910638046775629902

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