Properties

Label 10-3174e5-1.1-c1e5-0-1
Degree $10$
Conductor $3.221\times 10^{17}$
Sign $1$
Analytic cond. $1.04573\times 10^{7}$
Root an. cond. $5.03433$
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  + 5·2-s − 5·3-s + 15·4-s − 5-s − 25·6-s + 11·7-s + 35·8-s + 15·9-s − 5·10-s + 11·11-s − 75·12-s + 12·13-s + 55·14-s + 5·15-s + 70·16-s − 17-s + 75·18-s + 15·19-s − 15·20-s − 55·21-s + 55·22-s − 175·24-s − 9·25-s + 60·26-s − 35·27-s + 165·28-s + 29-s + ⋯
L(s)  = 1  + 3.53·2-s − 2.88·3-s + 15/2·4-s − 0.447·5-s − 10.2·6-s + 4.15·7-s + 12.3·8-s + 5·9-s − 1.58·10-s + 3.31·11-s − 21.6·12-s + 3.32·13-s + 14.6·14-s + 1.29·15-s + 35/2·16-s − 0.242·17-s + 17.6·18-s + 3.44·19-s − 3.35·20-s − 12.0·21-s + 11.7·22-s − 35.7·24-s − 9/5·25-s + 11.7·26-s − 6.73·27-s + 31.1·28-s + 0.185·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{5} \cdot 3^{5} \cdot 23^{10}\)
Sign: $1$
Analytic conductor: \(1.04573\times 10^{7}\)
Root analytic conductor: \(5.03433\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{5} \cdot 3^{5} \cdot 23^{10} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(207.4259163\)
\(L(\frac12)\) \(\approx\) \(207.4259163\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{5} \)
3$C_1$ \( ( 1 + T )^{5} \)
23 \( 1 \)
good5$C_2 \times (C_2^4 : C_5)$ \( 1 + T + 2 p T^{2} + 6 T^{3} + 28 T^{4} + 11 T^{5} + 28 p T^{6} + 6 p^{2} T^{7} + 2 p^{4} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \times (C_2^4 : C_5)$ \( 1 - 11 T + 68 T^{2} - 286 T^{3} + 136 p T^{4} - 2673 T^{5} + 136 p^{2} T^{6} - 286 p^{2} T^{7} + 68 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \times (C_2^4 : C_5)$ \( 1 - p T + 7 p T^{2} - 37 p T^{3} + 163 p T^{4} - 595 p T^{5} + 163 p^{2} T^{6} - 37 p^{3} T^{7} + 7 p^{4} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
13$C_2 \times (C_2^4 : C_5)$ \( 1 - 12 T + 94 T^{2} - 511 T^{3} + 2384 T^{4} - 9121 T^{5} + 2384 p T^{6} - 511 p^{2} T^{7} + 94 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \times (C_2^4 : C_5)$ \( 1 + T + 48 T^{2} + 21 T^{3} + 1325 T^{4} + 665 T^{5} + 1325 p T^{6} + 21 p^{2} T^{7} + 48 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \times (C_2^4 : C_5)$ \( 1 - 15 T + 174 T^{2} - 1322 T^{3} + 8298 T^{4} - 39473 T^{5} + 8298 p T^{6} - 1322 p^{2} T^{7} + 174 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \times (C_2^4 : C_5)$ \( 1 - T + 108 T^{2} - 47 T^{3} + 5205 T^{4} - 1067 T^{5} + 5205 p T^{6} - 47 p^{2} T^{7} + 108 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \times (C_2^4 : C_5)$ \( 1 + 18 T + 256 T^{2} + 2435 T^{3} + 19100 T^{4} + 116351 T^{5} + 19100 p T^{6} + 2435 p^{2} T^{7} + 256 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \times (C_2^4 : C_5)$ \( 1 - 10 T + 192 T^{2} - 1362 T^{3} + 389 p T^{4} - 73781 T^{5} + 389 p^{2} T^{6} - 1362 p^{2} T^{7} + 192 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \times (C_2^4 : C_5)$ \( 1 + 16 T + 248 T^{2} + 2381 T^{3} + 21158 T^{4} + 140791 T^{5} + 21158 p T^{6} + 2381 p^{2} T^{7} + 248 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \times (C_2^4 : C_5)$ \( 1 - 18 T + 228 T^{2} - 2287 T^{3} + 19450 T^{4} - 137245 T^{5} + 19450 p T^{6} - 2287 p^{2} T^{7} + 228 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \times (C_2^4 : C_5)$ \( 1 + 4 T + 138 T^{2} + 670 T^{3} + 11095 T^{4} + 40491 T^{5} + 11095 p T^{6} + 670 p^{2} T^{7} + 138 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \times (C_2^4 : C_5)$ \( 1 + T + 151 T^{2} + 66 T^{3} + 10429 T^{4} + 1907 T^{5} + 10429 p T^{6} + 66 p^{2} T^{7} + 151 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \times (C_2^4 : C_5)$ \( 1 - 2 T + 136 T^{2} - 448 T^{3} + 13205 T^{4} - 27631 T^{5} + 13205 p T^{6} - 448 p^{2} T^{7} + 136 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \times (C_2^4 : C_5)$ \( 1 + T + 136 T^{2} + 142 T^{3} + 8706 T^{4} + 13271 T^{5} + 8706 p T^{6} + 142 p^{2} T^{7} + 136 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \times (C_2^4 : C_5)$ \( 1 - 29 T + 568 T^{2} - 7447 T^{3} + 80941 T^{4} - 705107 T^{5} + 80941 p T^{6} - 7447 p^{2} T^{7} + 568 p^{3} T^{8} - 29 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \times (C_2^4 : C_5)$ \( 1 + 11 T + 245 T^{2} + 1661 T^{3} + 24131 T^{4} + 130053 T^{5} + 24131 p T^{6} + 1661 p^{2} T^{7} + 245 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \times (C_2^4 : C_5)$ \( 1 + 8 T + 120 T^{2} + 987 T^{3} + 13892 T^{4} + 76459 T^{5} + 13892 p T^{6} + 987 p^{2} T^{7} + 120 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \times (C_2^4 : C_5)$ \( 1 - 40 T + 958 T^{2} - 15945 T^{3} + 203176 T^{4} - 2024847 T^{5} + 203176 p T^{6} - 15945 p^{2} T^{7} + 958 p^{3} T^{8} - 40 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \times (C_2^4 : C_5)$ \( 1 - 8 T + 423 T^{2} - 2627 T^{3} + 70894 T^{4} - 325857 T^{5} + 70894 p T^{6} - 2627 p^{2} T^{7} + 423 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \times (C_2^4 : C_5)$ \( 1 - 2 T + 143 T^{2} - 853 T^{3} + 20492 T^{4} - 64973 T^{5} + 20492 p T^{6} - 853 p^{2} T^{7} + 143 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \times (C_2^4 : C_5)$ \( 1 - 17 T + 286 T^{2} - 3550 T^{3} + 37132 T^{4} - 371807 T^{5} + 37132 p T^{6} - 3550 p^{2} T^{7} + 286 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.10090520539402618731185514641, −4.98390902506183866717155665977, −4.92783476074500827060415491045, −4.66968881442184438520532032328, −4.59281041218632580755019914458, −4.30634250091168137008707925712, −4.26074983321649381191962452567, −4.12777972212021034289981633053, −3.95952470936422336089916876988, −3.70706040070686872532586181257, −3.51157378290755109839438703875, −3.42917223512504785106090752471, −3.42012626546148612207520117061, −3.38319761757124889788430098909, −2.95189047728993086517419247383, −2.09764865385811378511816036294, −2.02671707506926982232856938921, −1.95957502555241921865406795984, −1.91071301204308366097790381528, −1.88374512670597499121638051623, −1.29846118976158710537875400389, −1.11773203611966143124392243727, −1.07773873977739331198708438005, −0.901038056872679085861546948400, −0.845761601078044505215514377422, 0.845761601078044505215514377422, 0.901038056872679085861546948400, 1.07773873977739331198708438005, 1.11773203611966143124392243727, 1.29846118976158710537875400389, 1.88374512670597499121638051623, 1.91071301204308366097790381528, 1.95957502555241921865406795984, 2.02671707506926982232856938921, 2.09764865385811378511816036294, 2.95189047728993086517419247383, 3.38319761757124889788430098909, 3.42012626546148612207520117061, 3.42917223512504785106090752471, 3.51157378290755109839438703875, 3.70706040070686872532586181257, 3.95952470936422336089916876988, 4.12777972212021034289981633053, 4.26074983321649381191962452567, 4.30634250091168137008707925712, 4.59281041218632580755019914458, 4.66968881442184438520532032328, 4.92783476074500827060415491045, 4.98390902506183866717155665977, 5.10090520539402618731185514641

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