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Properties

Label 10-7360e5-1.1-c1e5-0-9
Degree $10$
Conductor $2.160\times 10^{19}$
Sign $-1$
Analytic cond. $7.01091\times 10^{8}$
Root an. cond. $7.66615$
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  − 3-s − 5·5-s + 7-s − 4·9-s − 3·11-s − 7·13-s + 5·15-s + 9·17-s − 19-s − 21-s − 5·23-s + 15·25-s + 4·27-s + 10·29-s + 21·31-s + 3·33-s − 5·35-s − 8·37-s + 7·39-s − 13·41-s + 6·43-s + 20·45-s − 5·49-s − 9·51-s + 6·53-s + 15·55-s + 57-s + ⋯
L(s)  = 1  − 0.577·3-s − 2.23·5-s + 0.377·7-s − 4/3·9-s − 0.904·11-s − 1.94·13-s + 1.29·15-s + 2.18·17-s − 0.229·19-s − 0.218·21-s − 1.04·23-s + 3·25-s + 0.769·27-s + 1.85·29-s + 3.77·31-s + 0.522·33-s − 0.845·35-s − 1.31·37-s + 1.12·39-s − 2.03·41-s + 0.914·43-s + 2.98·45-s − 5/7·49-s − 1.26·51-s + 0.824·53-s + 2.02·55-s + 0.132·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{30} \cdot 5^{5} \cdot 23^{5}\)
Sign: $-1$
Analytic conductor: \(7.01091\times 10^{8}\)
Root analytic conductor: \(7.66615\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{30} \cdot 5^{5} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{5} \)
23$C_1$ \( ( 1 + T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + T + 5 T^{2} + 5 T^{3} + 5 p T^{4} + 8 T^{5} + 5 p^{2} T^{6} + 5 p^{2} T^{7} + 5 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - T + 6 T^{2} + 24 T^{3} + 11 T^{4} + 178 T^{5} + 11 p T^{6} + 24 p^{2} T^{7} + 6 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 3 T + 6 T^{2} + 32 T^{3} + 219 T^{4} + 570 T^{5} + 219 p T^{6} + 32 p^{2} T^{7} + 6 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 7 T + 49 T^{2} + 185 T^{3} + 817 T^{4} + 2518 T^{5} + 817 p T^{6} + 185 p^{2} T^{7} + 49 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 9 T + 90 T^{2} - 2 p^{2} T^{3} + 3097 T^{4} - 14434 T^{5} + 3097 p T^{6} - 2 p^{4} T^{7} + 90 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + T + 20 T^{2} - 40 T^{3} + 669 T^{4} + 910 T^{5} + 669 p T^{6} - 40 p^{2} T^{7} + 20 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 10 T + 122 T^{2} - 596 T^{3} + 4337 T^{4} - 15484 T^{5} + 4337 p T^{6} - 596 p^{2} T^{7} + 122 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 21 T + 303 T^{2} - 2965 T^{3} + 23253 T^{4} - 142336 T^{5} + 23253 p T^{6} - 2965 p^{2} T^{7} + 303 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 8 T + 115 T^{2} + 948 T^{3} + 7744 T^{4} + 1240 p T^{5} + 7744 p T^{6} + 948 p^{2} T^{7} + 115 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 13 T + 181 T^{2} + 1655 T^{3} + 14447 T^{4} + 91910 T^{5} + 14447 p T^{6} + 1655 p^{2} T^{7} + 181 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 6 T + 13 T^{2} - 88 T^{3} + 2800 T^{4} - 19044 T^{5} + 2800 p T^{6} - 88 p^{2} T^{7} + 13 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 180 T^{2} + 50 T^{3} + 14819 T^{4} + 4764 T^{5} + 14819 p T^{6} + 50 p^{2} T^{7} + 180 p^{3} T^{8} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 6 T + 185 T^{2} - 856 T^{3} + 15418 T^{4} - 57060 T^{5} + 15418 p T^{6} - 856 p^{2} T^{7} + 185 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 18 T + 383 T^{2} + 4336 T^{3} + 50338 T^{4} + 386268 T^{5} + 50338 p T^{6} + 4336 p^{2} T^{7} + 383 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 11 T + 304 T^{2} - 2526 T^{3} + 37321 T^{4} - 226162 T^{5} + 37321 p T^{6} - 2526 p^{2} T^{7} + 304 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 38 T + 813 T^{2} + 11928 T^{3} + 134504 T^{4} + 1215844 T^{5} + 134504 p T^{6} + 11928 p^{2} T^{7} + 813 p^{3} T^{8} + 38 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 21 T + 485 T^{2} + 6153 T^{3} + 77909 T^{4} + 661928 T^{5} + 77909 p T^{6} + 6153 p^{2} T^{7} + 485 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 12 T + 366 T^{2} + 3120 T^{3} + 52589 T^{4} + 328336 T^{5} + 52589 p T^{6} + 3120 p^{2} T^{7} + 366 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 18 T + 305 T^{2} + 4256 T^{3} + 49304 T^{4} + 438044 T^{5} + 49304 p T^{6} + 4256 p^{2} T^{7} + 305 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 20 T + 427 T^{2} + 6048 T^{3} + 70342 T^{4} + 727384 T^{5} + 70342 p T^{6} + 6048 p^{2} T^{7} + 427 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 16 T + 295 T^{2} + 3324 T^{3} + 36264 T^{4} + 339288 T^{5} + 36264 p T^{6} + 3324 p^{2} T^{7} + 295 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 29 T + 742 T^{2} - 12014 T^{3} + 169335 T^{4} - 1785070 T^{5} + 169335 p T^{6} - 12014 p^{2} T^{7} + 742 p^{3} T^{8} - 29 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.90590623484730624439035236962, −4.81027566171375787456784108570, −4.61268388011206146921974854228, −4.50998749885264271459883558986, −4.48820923138682718976706978380, −4.47970549484393883931544585459, −4.22034477990801544761770100544, −4.00786121723702779469426664202, −3.73256099476610854908726656479, −3.68813208964859560518665759508, −3.23914779977114554770265087481, −3.22466158276533083770065070726, −3.17033281947353231456299148302, −3.02572999530785141896850307939, −2.86616207121364453900626204213, −2.61568085936233533541815599037, −2.60974333100114836473340189767, −2.50806471678826617676443481298, −2.05926238088964601160030758814, −2.03027706167142036152451372256, −1.51375623178117382283586561038, −1.27722080337919496522568413780, −1.12771983226195061090957751720, −1.12573979619492105489395744133, −0.987000469076901435101735103736, 0, 0, 0, 0, 0, 0.987000469076901435101735103736, 1.12573979619492105489395744133, 1.12771983226195061090957751720, 1.27722080337919496522568413780, 1.51375623178117382283586561038, 2.03027706167142036152451372256, 2.05926238088964601160030758814, 2.50806471678826617676443481298, 2.60974333100114836473340189767, 2.61568085936233533541815599037, 2.86616207121364453900626204213, 3.02572999530785141896850307939, 3.17033281947353231456299148302, 3.22466158276533083770065070726, 3.23914779977114554770265087481, 3.68813208964859560518665759508, 3.73256099476610854908726656479, 4.00786121723702779469426664202, 4.22034477990801544761770100544, 4.47970549484393883931544585459, 4.48820923138682718976706978380, 4.50998749885264271459883558986, 4.61268388011206146921974854228, 4.81027566171375787456784108570, 4.90590623484730624439035236962

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