Properties

Label 10-546e5-1.1-c7e5-0-5
Degree $10$
Conductor $4.852\times 10^{13}$
Sign $-1$
Analytic cond. $1.44349\times 10^{11}$
Root an. cond. $13.0599$
Motivic weight $7$
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  − 40·2-s + 135·3-s + 960·4-s − 340·5-s − 5.40e3·6-s + 1.71e3·7-s − 1.79e4·8-s + 1.09e4·9-s + 1.36e4·10-s − 1.30e3·11-s + 1.29e5·12-s + 1.09e4·13-s − 6.86e4·14-s − 4.59e4·15-s + 2.86e5·16-s − 4.24e3·17-s − 4.37e5·18-s − 1.69e4·19-s − 3.26e5·20-s + 2.31e5·21-s + 5.21e4·22-s − 7.80e4·23-s − 2.41e6·24-s − 1.77e5·25-s − 4.39e5·26-s + 6.88e5·27-s + 1.64e6·28-s + ⋯
L(s)  = 1  − 3.53·2-s + 2.88·3-s + 15/2·4-s − 1.21·5-s − 10.2·6-s + 1.88·7-s − 12.3·8-s + 5·9-s + 4.30·10-s − 0.295·11-s + 21.6·12-s + 1.38·13-s − 6.68·14-s − 3.51·15-s + 35/2·16-s − 0.209·17-s − 17.6·18-s − 0.568·19-s − 9.12·20-s + 5.45·21-s + 1.04·22-s − 1.33·23-s − 35.7·24-s − 2.26·25-s − 4.90·26-s + 6.73·27-s + 14.1·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 7^{5} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 7^{5} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{5} \cdot 3^{5} \cdot 7^{5} \cdot 13^{5}\)
Sign: $-1$
Analytic conductor: \(1.44349\times 10^{11}\)
Root analytic conductor: \(13.0599\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{546} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{5} \cdot 3^{5} \cdot 7^{5} \cdot 13^{5} ,\ ( \ : 7/2, 7/2, 7/2, 7/2, 7/2 ),\ -1 )\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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$C_1$ \( ( 1 + p^{3} T )^{5} \)
3$C_1$ \( ( 1 - p^{3} T )^{5} \)
7$C_1$ \( ( 1 - p^{3} T )^{5} \)
13$C_1$ \( ( 1 - p^{3} T )^{5} \)
good5$C_2 \wr S_5$ \( 1 + 68 p T + 58578 p T^{2} + 608986 p^{3} T^{3} + 1444916821 p^{2} T^{4} + 61790108436 p^{3} T^{5} + 1444916821 p^{9} T^{6} + 608986 p^{17} T^{7} + 58578 p^{22} T^{8} + 68 p^{29} T^{9} + p^{35} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 1303 T + 42108237 T^{2} - 24597842260 T^{3} + 916878414481288 T^{4} - 1475553724993041462 T^{5} + 916878414481288 p^{7} T^{6} - 24597842260 p^{14} T^{7} + 42108237 p^{21} T^{8} + 1303 p^{28} T^{9} + p^{35} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 4247 T + 1613690547 T^{2} + 8149276487656 T^{3} + 1156582243463969020 T^{4} + \)\(52\!\cdots\!66\)\( T^{5} + 1156582243463969020 p^{7} T^{6} + 8149276487656 p^{14} T^{7} + 1613690547 p^{21} T^{8} + 4247 p^{28} T^{9} + p^{35} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 16984 T + 83138398 p T^{2} + 24210102187460 T^{3} + 2225069258269514477 T^{4} + \)\(31\!\cdots\!08\)\( T^{5} + 2225069258269514477 p^{7} T^{6} + 24210102187460 p^{14} T^{7} + 83138398 p^{22} T^{8} + 16984 p^{28} T^{9} + p^{35} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 78072 T + 13572347822 T^{2} + 650463352125228 T^{3} + 71145272160697806769 T^{4} + \)\(25\!\cdots\!72\)\( T^{5} + 71145272160697806769 p^{7} T^{6} + 650463352125228 p^{14} T^{7} + 13572347822 p^{21} T^{8} + 78072 p^{28} T^{9} + p^{35} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 213142 T + 64225137588 T^{2} + 8938676973052754 T^{3} + 59267307802432797791 p T^{4} + \)\(18\!\cdots\!96\)\( T^{5} + 59267307802432797791 p^{8} T^{6} + 8938676973052754 p^{14} T^{7} + 64225137588 p^{21} T^{8} + 213142 p^{28} T^{9} + p^{35} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 186027 T + 103628303579 T^{2} + 15060090909010996 T^{3} + \)\(49\!\cdots\!38\)\( T^{4} + \)\(57\!\cdots\!18\)\( T^{5} + \)\(49\!\cdots\!38\)\( p^{7} T^{6} + 15060090909010996 p^{14} T^{7} + 103628303579 p^{21} T^{8} + 186027 p^{28} T^{9} + p^{35} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 101025 T + 252470877075 T^{2} - 9467060729277200 T^{3} + \)\(34\!\cdots\!80\)\( T^{4} - \)\(90\!\cdots\!50\)\( T^{5} + \)\(34\!\cdots\!80\)\( p^{7} T^{6} - 9467060729277200 p^{14} T^{7} + 252470877075 p^{21} T^{8} - 101025 p^{28} T^{9} + p^{35} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 23976 T + 545534485325 T^{2} + 514334524827024 p T^{3} + \)\(15\!\cdots\!70\)\( T^{4} + \)\(54\!\cdots\!36\)\( T^{5} + \)\(15\!\cdots\!70\)\( p^{7} T^{6} + 514334524827024 p^{15} T^{7} + 545534485325 p^{21} T^{8} + 23976 p^{28} T^{9} + p^{35} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 55528 T + 673914196698 T^{2} + 141772770098277572 T^{3} + \)\(26\!\cdots\!81\)\( T^{4} + \)\(54\!\cdots\!16\)\( T^{5} + \)\(26\!\cdots\!81\)\( p^{7} T^{6} + 141772770098277572 p^{14} T^{7} + 673914196698 p^{21} T^{8} + 55528 p^{28} T^{9} + p^{35} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 985981 T + 1276446419051 T^{2} + 624419720944947068 T^{3} + \)\(52\!\cdots\!70\)\( T^{4} + \)\(20\!\cdots\!90\)\( T^{5} + \)\(52\!\cdots\!70\)\( p^{7} T^{6} + 624419720944947068 p^{14} T^{7} + 1276446419051 p^{21} T^{8} + 985981 p^{28} T^{9} + p^{35} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 1891657 T + 3098769350825 T^{2} + 3032598338798101916 T^{3} + \)\(52\!\cdots\!30\)\( T^{4} + \)\(55\!\cdots\!18\)\( T^{5} + \)\(52\!\cdots\!30\)\( p^{7} T^{6} + 3032598338798101916 p^{14} T^{7} + 3098769350825 p^{21} T^{8} + 1891657 p^{28} T^{9} + p^{35} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 2802208 T + 8769236776287 T^{2} + 18662528794625869616 T^{3} + \)\(38\!\cdots\!46\)\( T^{4} + \)\(60\!\cdots\!76\)\( T^{5} + \)\(38\!\cdots\!46\)\( p^{7} T^{6} + 18662528794625869616 p^{14} T^{7} + 8769236776287 p^{21} T^{8} + 2802208 p^{28} T^{9} + p^{35} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 1140591 T + 4932699775605 T^{2} - 11864523329163884 T^{3} + \)\(25\!\cdots\!10\)\( T^{4} - \)\(30\!\cdots\!06\)\( p T^{5} + \)\(25\!\cdots\!10\)\( p^{7} T^{6} - 11864523329163884 p^{14} T^{7} + 4932699775605 p^{21} T^{8} - 1140591 p^{28} T^{9} + p^{35} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 265168 T + 24241180772395 T^{2} - 11578680895063746576 T^{3} + \)\(25\!\cdots\!10\)\( T^{4} - \)\(12\!\cdots\!52\)\( T^{5} + \)\(25\!\cdots\!10\)\( p^{7} T^{6} - 11578680895063746576 p^{14} T^{7} + 24241180772395 p^{21} T^{8} - 265168 p^{28} T^{9} + p^{35} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 4483276 T + 39774202712459 T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(64\!\cdots\!46\)\( T^{4} + \)\(16\!\cdots\!80\)\( T^{5} + \)\(64\!\cdots\!46\)\( p^{7} T^{6} + \)\(12\!\cdots\!80\)\( p^{14} T^{7} + 39774202712459 p^{21} T^{8} + 4483276 p^{28} T^{9} + p^{35} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 2350578 T + 31153469599224 T^{2} + 91122815818795101526 T^{3} + \)\(50\!\cdots\!35\)\( T^{4} + \)\(14\!\cdots\!80\)\( T^{5} + \)\(50\!\cdots\!35\)\( p^{7} T^{6} + 91122815818795101526 p^{14} T^{7} + 31153469599224 p^{21} T^{8} + 2350578 p^{28} T^{9} + p^{35} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 4079889 T + 631965451299 p T^{2} + \)\(12\!\cdots\!68\)\( T^{3} + \)\(15\!\cdots\!88\)\( T^{4} + \)\(38\!\cdots\!06\)\( T^{5} + \)\(15\!\cdots\!88\)\( p^{7} T^{6} + \)\(12\!\cdots\!68\)\( p^{14} T^{7} + 631965451299 p^{22} T^{8} + 4079889 p^{28} T^{9} + p^{35} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 8731571 T + 105915930047289 T^{2} + \)\(58\!\cdots\!24\)\( T^{3} + \)\(42\!\cdots\!68\)\( T^{4} + \)\(19\!\cdots\!10\)\( T^{5} + \)\(42\!\cdots\!68\)\( p^{7} T^{6} + \)\(58\!\cdots\!24\)\( p^{14} T^{7} + 105915930047289 p^{21} T^{8} + 8731571 p^{28} T^{9} + p^{35} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 20077879 T + 366698441511765 T^{2} + \)\(40\!\cdots\!04\)\( T^{3} + \)\(39\!\cdots\!50\)\( T^{4} + \)\(27\!\cdots\!54\)\( T^{5} + \)\(39\!\cdots\!50\)\( p^{7} T^{6} + \)\(40\!\cdots\!04\)\( p^{14} T^{7} + 366698441511765 p^{21} T^{8} + 20077879 p^{28} T^{9} + p^{35} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 3780209 T + 1536217199893 T^{2} + \)\(63\!\cdots\!36\)\( T^{3} + \)\(96\!\cdots\!94\)\( T^{4} - \)\(41\!\cdots\!14\)\( T^{5} + \)\(96\!\cdots\!94\)\( p^{7} T^{6} + \)\(63\!\cdots\!36\)\( p^{14} T^{7} + 1536217199893 p^{21} T^{8} - 3780209 p^{28} T^{9} + p^{35} T^{10} \)
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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

−5.78115909057925946865286599320, −5.77390134278348588893507215249, −5.70790499545174414802055932610, −5.60047519578141886260287134500, −5.49827375629335565081276702781, −4.55404309607707753848088365467, −4.45047958381446750722661393550, −4.36611303493675314780663792460, −4.35220980689704605313968489856, −4.16014137942635234413039605055, −3.58326779617546502788000512529, −3.36878125568649746244023792179, −3.35520150845133289606417547551, −3.25669769505401763524987391694, −3.17302582209710287068071380597, −2.42065464823011514864294167212, −2.32034271108852319783765651775, −2.14776007743985102504299177008, −2.06179092351260024774526926577, −2.05972447667566170114915713611, −1.51971075104965164580197530872, −1.33916050273644356725586202753, −1.32114274971617126046260370601, −1.18463145882928958294761094077, −1.00538008438317754033556133975, 0, 0, 0, 0, 0, 1.00538008438317754033556133975, 1.18463145882928958294761094077, 1.32114274971617126046260370601, 1.33916050273644356725586202753, 1.51971075104965164580197530872, 2.05972447667566170114915713611, 2.06179092351260024774526926577, 2.14776007743985102504299177008, 2.32034271108852319783765651775, 2.42065464823011514864294167212, 3.17302582209710287068071380597, 3.25669769505401763524987391694, 3.35520150845133289606417547551, 3.36878125568649746244023792179, 3.58326779617546502788000512529, 4.16014137942635234413039605055, 4.35220980689704605313968489856, 4.36611303493675314780663792460, 4.45047958381446750722661393550, 4.55404309607707753848088365467, 5.49827375629335565081276702781, 5.60047519578141886260287134500, 5.70790499545174414802055932610, 5.77390134278348588893507215249, 5.78115909057925946865286599320

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